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Journal of the American 
Oriental Society 

American Oriental Society 



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UNIVERSITY OF CALIFORNIA. 



GIFT OF 



DANIEL C. OILMAN. 



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JOURNAL 



OF THE 



AMERICAN ORIENTAL SOCIETY. 



SIXTH VOLUME. 




NEW HAVEN: 

FOR THE AMERICAN ORIENTAL SOCIETY, 

pRurraD bt E. Hayw, Prottbr to Yau Coubol 

MDOOOLX. 



SOLD BT THE BOODRT'S AGDflTB: 

NEW YORK: JOHN WILEY, 66 WALKER ST.; 

LONDON: TRUBNER <fc CO.; PARIS: BEN J. DUPRAT;' 

LEIPZIG: F. A. BROCKHAUS. 



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COMMITTEE OF PUBLICATION 

or THE 

AMERICA* ORIENTAL SOCIETY, 

Fob thb years 1868-60. 



Edward £. Salisbury, New Haven. 
William D. Whitney, m 

Jambs Hadlby, u 

Ezra Abbot, Cambridge* 

William W. Turner, Washington. 



h ^xm»vw^>»|i>^M*iw^»WW^»0^* W ^i 



Entered eeoordlnf to Act el Confrem, In the year I860, by the 

Amsrioan Oriental Sooixtt, 

in the Clerk's Office of the Dietriet Coert ef CojueetiMit. 



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CONTENTS 



SIXTH VOLUME. 



Asr. L — Ahaltsb and Extracts of X*&£vJt o!j** V^» J* 00 * °» 
IBs Balance of Wisdom, a* Ababio Woes ok thi Watrr- 
salahob, w ri tt en bt 'AL-KalraJ m the Twelfth Century. 
By the Chevalier N. Khanisoff, Russian Consul General at Tabris, 
Persia, 1 

Aai. IL— ObSEBYATIOHB OH THE PREPOSITIONS, OoKJUNOTIOBB, AMD OTHSE 
FaETIOLBB OF THE IbIEULU AMD RE CoOEATB LANGUAGES. By ROT. 

Lewis Grout, Ha w o u ar y of the A. B. 0. F. M. in South Africa, 129 

Aet. HX — Teamblation op the Stf bta-SiddhInta, a Text-Book of Hindu 
Aeteoeomt ; with Notes, and an Appendix, By Rer. Ebeneesr 
Burgess, formerly Missionary of the A. B. 0. F. M. in India, 
assisted by the Committee of Publication, 141 

Aet. IV. — Two Sanskrit Inscriptions, engraven oh stohb: the Original 
Texts, with Translations and Comments. By Frs-Edward Hall, 
Esq.,M.A^ 499 

Aet. V.— These Sanskrit Inscriptions, relating to Gbahtb of Land: 
the Original Texts, Translations, and Notes. By Fits-Edward 
Hall, Esq., M. A^ 688 

Aet. VX— A Greek Inscription from Daphne, near Ahtiooh, ih Stria. 

By James Hadlet, Professor of Greek in Tale College, 650 

Aet. VII. — Oh the Aeta-Siddhahta. By Fite-Edwaed Hall, Esq., M. A. 
[with an Additional Note on Aryabbatta and his Writings, by the 
Committee of Publication], 656 



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11 

Ml80ELLANIB8 I - 

roge. 

I Inverted Construction of Modern Armenian. By Rev. Euab RiGQft, D.D, 565 

\^ IX On Dr. 8. W. WilliamSt Chine* Dictionary. By Rev. William A. 

Maot 566 

III. On the Natural Limit* of Ancient Oriental History. By Prot James 

Moffat, D.D, ., 571 

IV. Extracts fbom Couubpondsicb : 

1. From a Letter of Rev. Justin Iferkins, D. D n of Orumiah, 574 

2. From a Letter of Prof. C. J. Thrnberg, of the University of Lund, 574 

3. From a Letter of Baja Bddhdkdnta Deva Bahddur, of Calcutta, 575 

4. From a Letter of John Muir, Esq., D.CJL., of Edinburgh (to F. E. 

Halt, Esq.), 576 

American Oriental Society: 

I. Select Minutes of Meetings of the Society, 577 

IX Additions to the Library and Cabinet, October, 1856—J/ay, 1860, ... 588 

III. List of Members, May, 1860, 607 



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ARTICLE I. 



ANALYSIS AND EXTRACTS 



* e X 4? ^Ijl-a^o ±>LjLS 



BOOK OF THE BALANCE OF WISDOM, 
AN ARABIC WORK ON THE WATER-BALANCE, 

WRITTEN BY 'AL-KHAZINI IN THE TWELFTH CRNTURT. 

By the Chevalier N. KHANIKOFF, 

AT TABRfl, rRRfllA. 



Presented to the Society October 29, 1857. 



[Our correspondent having communicated his paper to as in the 
French language, accompanied with the extracts in the original Arabic, 
we have taken the liberty to put it into English, and have in fact re- 
translated the extracts rather than give them through the medium of the 
French version. M. KhanikofTs own notes are printed on the pages to 
which they refer. To these we have added others, relating to the original 
text and its contents, which are distinguished by letters and numerals, 
and will be found at the end of the article. — Comm. of Publ.] 



The scantiness of the data which we possess for appreciating 
the results arrived at by the ancient civilizations which preceded 
that of Greece and Borne, renders it impossible for us to form 
any probable conjecture respecting the development which our 

S resent knowledge might have attained, if the tradition of the 
iscoveries made by the past in the domain of science had been 
transmitted without interruption, from generation to generation, 
down to the present time. But the history of the sciences pre- 
sents to us, in my opinion, an incontestable fact of deep signifi- 
cance: the rediscovery, namely, in modern times, of truths 
laboriously established of old ; and this fact is of itself enough 
to indicate the necessity of searching carefully in the scientific 
heritage of the past after all that it may be able to furnish us 
for the increase of our actual knowledge ; for a double diseov- 



VOL. VI. 



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2 N. Khanikoff, 

ery, necessarily requiring a double effort of human intellect, is 
an evident waste of that creative force which causes the advance 
of humanity in the glorious path of civilization. Modern orien- 
talists are tleginning to feel deeply the iustice and the importance 
of the counsel given them by the author of the M^canique Ce- 
leste, who, in his Compendium of the History of Astronomy, 
while persuading them to extract from the numerous oriental 
manuscripts preserved in our libraries whatever they contain 
that is or value to this science, remarks that " the grand varia- 
tions in the theory of the system of the world are not less inter- 
esting than the revolutions of empires ;" and the labors of MM. 
Chezy, Stanislas Julien, Am. S&lillot, Woepcke, Bochart, Spren- 
gel, Moreley, Dorn, Clement-Mullet, and others, have enriched 
with a mass of new and instructive facts our knowledge re- 
specting the state of the sciences in the Orient. Notwithstand- 
ing this, however, it must be granted that M. Clement-Mullet 
was perfectly justified in saying, as he has done in an article on 
the Arachnids, published in the Journal Asiatique,* that re- 
searches into the physical sciences of the Orientals have been 
entirely, or almost entirely, neglected ; and it is only necessary 
to read the eloquent pages in which the author of the Cosmos 
estimates the influence of the Arab element upon European civ- 
ilization, to be convinced of the scantiness of our information 
as to the condition of physical science among the Arabs ; for 
that illustrious representative of modern civilization, after hav- 
ing shown that the Arabs had raised themselves to the third step 
in the progressive knowledge of physical facts, a step entirely 
unknown to the ancients, that, namely, of experimentation, con- 
cludes that,f " as instances of the progress which physical sci- 
ence owes to the Arabs, one .can only mention the labors of 
Alhazen respecting the refraction of light, derived perhaps in 
part from the Optics of Ptolemy, and the discovery and first 
application of the pendulum as a measure of time, by the great 
astronomer Ebn-Yunis." 

All this leads me to suppose that men of science will be inter- 
ested to have their attention called to a work of the twelfth cen- 
tury, written in Arabic, which treats exclusively of the balance, 
and of the results arrived at by the help of that instrument, 
which has given to modern science so many beautiful discoveries. 
I hesitated for some time whether to offer a pure and simple 
translation of this work, or a detailed analysis of its contents, 
presenting in full only those passages which contain remarkable 
matter, worthy of being cited. Finally, taking into considera- 
tion the numerous repetitions, the superfluity of detail, and even 

* Number for Aug.— Sept, 1854, 5«e Bene, iv. 214, etc. 
f Kosmos, ii. 258 (original edition). 



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Book of Hie Balance of Wisdom. 3 

the obscurity of exposition, of matters which, thanks to the pro- 
gress of science, have become for us elementary, and which, if 
presented in the little attractive form of the original text, would 
tend rather to conceal than to develop the interesting facts which 
it contains, I have decided to translate, in full, only the preface 
and introduction of the work, its exposition of the principles 
of centres of gravity, and its researches into the specific gravi- 
ties of metals, precious stones, and liquids, and to limit myself, 
beyond this, to citing the words of the author as pikes justifica- 
Hues, to show whether I have fully apprehended the sense of his 
reasonings. 

I have had at my disposal only a single manuscript copy of 
this work, which moreover lacks a few leaves in the middle and 
at the end, so that it has been impossible to determine its age : 
to judge from the chirography, however, it is quite ancient, and 
the absence of diacritical points sufficiently indicates that it is a 
work of the scribes of Ispahan, who have the bad habit of omit- 
ting these points, so essential to the correct reading of oriental 
texts. The original of each extract, whether longer or shorter, 
will be found accompanying its translation. It only remains for 
me to say that I have been scrupulous to render as faithfully 
as possible the text of my author, wherever I have cited from 
him ; in the cases where 1 have had to fill out the ellipses so 
common in Arabic, I have marked the words added by placing 
them in brackets. 

The work commences thus : 

Jy* ^fcualt s\+^ Lu*i ijata^} JjmlJ tOLx: J>l *{**} | ^-k>JI jJI**^ 

Jjtii!l 3 f£&* Jj* 3 jlli' jjlc UkxJIj qJOJI ^sja&y J**^ fW ^^ vJ 

In the name of God, the Compassionate, the Merciful. 

Praise be to God, beside whom there is no deity, the Wise, the True, 
the Just ! and may the blessing of God rest upon all His prophets and 
ambassadors, whom He has sent to His servants in order to justice, 
singling out our Prophet Muhammad, the Elect, to be the bearer of the 
law mild in righteousness ! 

Now, then, to our subject. Justice is the stay of all virtues, and the 
support of ail excellencies. For perfect virtue, which is wisdom in its 
two parts, knowledge and action, and in its two aspects, religion and the 



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4 K Khanikoff, 

^U-Xii^ JUXJ! *L**Jji Ls^Xaamo JUJi .Lo «jj^ JJuJt ajJj toLfi \jJ^> 
^JLJt &JLfc *^ib 8,Ltt! ^ 3 r LkJl i v ya3ft jufl J* y>^3 

JU)*3t &o Jl «tX^j Jc\*Jl <jo&>t UJ3 (j*Oftj O^^Jt vi>wol5 J*Aa!L» 
m>Lfc v-jji3 ^t **>-^ *»^ *Lto-H *k> ftJLe dJt (jtoldt iuJUJi aJjuJ^ 

OjU© *Xa> ^ ft*/* ^ ^M^ ^^ ft*' 6 o^ fl*™* ^Ji^+t a ***/?^ 
^ ^ LT^ a K U Jf ^ jjla ^yuii ^ «5JJJ3 »^ Jj^ 

Us>l3 iU^I ^3 ^aJ^I a ^>3j vJljMt i **!^ L*fll *La*t ^Ui" 

course of the world, consists of perfect knowledge and assured action ; 
and justice brings the two [requisites] together. It is the confluence of 
the two perfections of that virtue, the means of reaching the limits of all 
greatness, and the cause of securing the prize* in all excellence. In order 
to place justice on the pinnacle of perfection, the Supreme Creator made 
Himself known to the Choicest of His servants under the name of the 
Just ; and it was by the light of justice that the world became complete 
and perfected, and was brought to perfect order — to which there is 
allusion in the words of the Blessed : " by justice were the heavens and 
the earth established ;" and, having appropriated to justice this elevated 
rank and lofty place, God has lavished upon it the robes of complacency 
and love, and made it an object of love to the hearts of all His servants ; 
so that human nature is fond of it, and the souls of men yearn after it, 
and may be seen to covet the experience of it, using all diligence to 
secure it If any thing happens to divert men from it, or to incline 
them to its opposite, still they find within themselves a recognition of it 
and a confirmation of its real nature ; so that the tyrant commends the 
justice of others. For this reason, also, one sees the souls of men 
pained at any composition of parts which is not symmetrical, and so 
abhorring lameness and blindness, and auguring ill therefrom. More- 
over, in order to the preservation of the empire of justice, the Supreme 

* vJU«Jf v ^aaqS L e. reed of precedence. By this name is designated a lance 
planted in the middle of a plain, where horse -races are held, and which the leader 
in the nice aeiies in passing. 



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Booh of ike Balance of Wisdom. 5 

*4aU a+^JU Stolstj j^j ^x^ULJ bO>y \y>\j&>»\ J all! Lo ; tjJiS a jJ^\ 

v\>i rOU "i ^y^- XSaS^JI ^Lft *gJwU l5 ^;^ AaJl^JI ^^-O UX>- &A*>) 

^UJI oiA^a^P (JUft £ JcX*J! 3 jJUJt J. Jjutlt o- g^j "tf &>La«Jt 
!y J^jJI £ J^^ »LaA^ <*U>Jl A4J^ ^ b^bt WU.jiaj i^s^ ^^ 

Ji> 5-ac .^c «t«3! \J^*> *iy> j^-o J&oJt v£*J fefll+j (21^5 8 1>^ CT* 

Jub JUJL& { y XaJ U^ J %,! *J u£***J Q4i ^^^3 »r*>^ &>l**J V ^^D Jt^ 

God lias made the side-members of man's body in pairs, and its middle 
members single ; and He calls men to pursue the paths of felicity by 
the practice of justice, and adherence to uprightness, according to the 
dffine words : " and do justly — verily, God loves them who do justly,"* 
and again : " as for them who say 'our Lord is God,' and are upright"! — 
wishing to do them good, and lavishing mercy upon them. God has 
even set up justice as the criterion of judgment between His creatures, 
being content with equity ; so that no one will pass the bridge of salva- 
tion without a certificate of uprightness in action, nor repose in the 
paradise of felicity without a diploma of justice in knowledge. 

Justice in knowledge is the verification of the object of knowledge in 
accordance with its scope, in the way appropriate to it, kept clear of 
the defect of doubt and uncertainty. Justice in action is two-fold : 
1. self-government, which is the harmonizing of the natural endow- 
ments, the maintenance of equilibrium between the powers of the soul, 
and the bringing of them under beautiful control — agreeably to the 
aying : ** the most just of men is he who lets his reason arbitrate for 
hk desire ;" and it is a part of the perfection of such a man that he 
dispenses justice among those inferior to himself and wards off from 
others any injury whith he has experienced, so that men are secure as 
to his doing evil ; 2. control over others, which consists in the main- 
tenance of moderation within one's self, together with a power of con- 
rtraint, in respect to the performance of obligations to men, and the 
requiring of that performance on their part. 

• Kur. xhx 9. f Kur. xlvi 12. 



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6 N. Khamkoff, 

ajLj* ^bu jJJf )U&) SsjUjJj la! pLaiif } ^JSjJI H 3/ jJL w&**x~t 
^,yj JiAjJJ ^ylu q! ^V,l iO(Jc*m g^S ^Ifc ^jySjf^ 8s>Lx; ^JLa* 

lyl^ v^A^t xj^ j*nPjJ^ fgrUb yC>ty ^LhV ffr" ^ Jjjlk ^1 
r-s^J! f»Ltai &JUJ>) aju^ au$L J ^ ^o f'6A^ r ^^ W^ ^ vJl>>S 

xsts ^ oj^. >* J^x*» f«4* 0-*^**^ ^ r^ r*** ^^ o 1 * 

d^t&JI u»l^t J^fi sl^ufl J^** ijft ua^OI hUjJ^ *xJt s^>s=U 
G LLLJI jJUJJ *J^ *Jyb *Jt ^UJ» Jj^JI ^1 jfe* olfcAll 3 
\pJK 5jSI ^IaJIj pLh* Jf *Jt ^L jy^t A ^ibu *IR 5A 

Justice is the support of both religion and the course of the world, 
and the stay of future as well as present felicity ; so that whoever takes 
hold of it, or of one of its branches, takes hold of a strong handle to 
which there is no breaking. Furthermore, because the mercy of the 
Supreme God intended to secure the rewards of virtue to His servants, 
and to establish them in the open way of His rectitude, He willed that 
justice should abide among them to the last day, uninterrupted, and 
unimpaired by the lapse of times and aces. Knowing that men would 
injure one another by compliance with the requirements of their natural 
impulses, He gave them self-command, as an inherent prerogative of 
their being — which they are naturally capable of and fitted for — and, 
in the amplitude of His mercy, and the breadth of His compassion, has 
provided for them, with constant goodness, by raising up among them 
just judges, their never-failing securities for justice. Of these there are 
three, answering to the several divisions of justice : 1. the glorious 
Book of God, which, from beginning to end, is without any admixture 
of error, is the supreme canon, to which both legal rules and doctrinal 
principles refer back, the arbiter between the Supereminent and the 
subject creature, to which the tradition of the Blessed Prophet is the 
sequel ; 2. the guided leaders and established doctors, set up in order 
to the dissipation of uncertainties and the removal of doubts, who 
are the vicars of the Prophet, and his substitutes, in every age and 
time, who protect the way of religion, and guide men into the paths of 



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Book of the Balance of Wisdom. 7 

<j JtX«i^ f»l£ijJ! ^ »; .J r JtjiL^Jt ^j*Uyai> J«m3a! juuliuJj JUjI 

o'j*^ c 5 ^ ^ *w^vy^3 J* jl& *u«Jt ^ i *^ *v* ij*Jt 

^ J«% oli^ 1 'W 5 ^ -k«J* a^' Wi> o!^ 1 i l^ 1 ' * a' 

Cjtjftlt* JU ^rfOAi H^ j^JoJ^ j^L^O KmJUl j^u U jjjdl mua> a K J^UIj 

felicity, when attacked by doubts and uncertainties ; of whom is the just 
ruler alluded to in the words of the Blessed : " the Sultan is the shadow 
of the Supreme God, upon His earth, the refuge of every one injured, 
and the judge ;" 3. the balance, which is the tongue of justice, the 
article of mediation between the commonalty and the great ; the crite- 
terion of just judgment, which with its final decision satisfies all the 
good and wicked, just-doers and doers of iniquity ; the standard, by its 
rectitude, for the settlement of men's altercations ; the security of order 
and justice among men, in respect to things which are left to them and 
committed to their disposal ; made by God the associate of His Kuran, 
which He joins on to the pearl-strings of His beneficence, so that 
the Supreme says : " God who hath sent down the True Book and the 
balance,"* and connects the benefit of the institution of the balance 
with that of the raising of the heavens, in the divine words : " and as 
for the heavens, He hath raised them; and He hath instituted the 
balance ; transgress not respecting the balance, do justice in weighing, 
and diminish not with the balance."! The Supreme God also says : 
" weigh with an even balance"! Indeed, the balance is one of the Su- 
preme God's lights, which He nas bountifully bestowed upon His serv- 
ants, out of the perfection of His justice, in order that they may thereby 
distinguish between the true and the false, the right and the wrong. 
For the essence of light is its being manifest of itself and so seen, and 
that it makes other things manifest, and is thus seen by ; while the 
balance is an instrument which, of itself, declares' its own evenness or 
deflection, and is the means of recognizing the. rectitude or deviation 
therefrom of other things. It is on account of the great power of the 
balance, and its binding authority, that God has magnified and exalted 

* Kor. adii. 16. f £ur. * y - fl » '. 8 - t ?ur. xvu. 87. 



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8 JV. KJiarukoff, 

njLo *j ki>Ju- v£*£> s^oi ^s^^ *it£ aJJI Jic U s^ot *^3£ *i>^ i^X^-^ 
Jfi»»a*L (_#«Jul |^ily olrft-^2 v-A*^' ^*$ WA ^f*!$ v>^" J^& v^a-ywjt^ 

-siL fcftLtf b^"it jot y> tjxil lib juja 4 2^0 jujcii uijil 3 
Q^y* JoutSi j^-^ x^mLaXI *^-&2 >Jl»Ji fly* ** ^£^ J*>j«j! -^m ^ 

Lyti' Lc> ji^i t\fe Ja^fiiL QJjftit j^t ^ Ui ^jmm jJiu ibl ^U? 

vJilkU jj*l' y ^ ^*J <3)^ J* & *jeLU^ iUX^l \jka Jutji oljuu 3, 
^xlae ^jl^Uuo^ *jf&2l\ aJdUil^ jl&'fl Juua-yL&i i^Ait iUXsM olj** 

qjjJ! aa> d L ^V!W L^ cU*aK ^IcX^ ^ ^ouJ^ £» W cr **• ^ o^^ 

it to such a degree as to rank it with His Book and the sword, in the 
divine words : " and we have sent down, with them, the Book ; and 
the balance, in order that men may do justice ; and we have sent down 
the cutting blade, which has exceeding force."* So, then, the balance 
is one of the three supports of that justice by which the world stands ; 
and justice is called "God's balance among His servants," both on 
account of this relationship of the balance thereto, and because it 
typifies the justice of the last day, clear of all injustice in its sentence, 
which is signified by the words of the Supreme God : " shall not, then, 
a soul be wronged at all ?"f and " he to whom the balance with just 
measure shall be given, will have much good done to him ; and none 
are mindful thereof save those who have hearts."! 

Sect. 1. Enumeration of the Advantages and Uses of the Balance of 

Wisdom. 
Says 'al-Kh&zint, after speaking of the balance in general, — The bal- 
ance of wisdom is something worked out by human intellect, and perfected 
by experiment and trial, of great importance on account of its advantages, 
and because it supplies the place of ingenious mechanicians. Among 
these [advantages] are : 1. exactness in weighing : this balance shows 
variation to the extent of a mithkAl, or of a grain, although the entire 
weight is a thousand mithk&ls, provided the maker has a delicate hand, 
attends to the minute details of the mechanism, and understands it ; 

* Kur. lvii 26. f ?«*• xxl 48, with bbt for bU . 

\ Kur. u. 272, altered. 



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Book of ilie Balance of Wisdom. 



o ( j** cr c^ <y** ^'j^ o* r>' ^>j&i g^' -*j?^ J L* ** 

qjPIj oS; f r*"v *AA^ /AA** 3>' L>°t^-^* .j' ^)u^o (jiaJU (j* lutein g^b 
^y^t til *UI £ y>ft J^ o 1 ^' ***' Oj* J*** ** ^J** ***IA L?^ 

D^>" L^' /^^ " ^J** JUwtliij U#i U^ ^ <ja*J ^t Lgdaa* 
CWjjl^' ,T?~* v-^X3l cW Jyutf II Ifrt ^*)!>M j!>L*« LibL^U aOij q* 

*j£ ^£\j Xaa^ f-Lyil Oyu jAjljjJtj *PLu\U S^"*!^ **M*J|$ jAm^^ 

t^ujj *j't3 *ytf ^^cXJt j^» .J! «LmSci oL^u^it Xla**^ -a£ iy f-L-i'il 

2. that it distinguishes pure metal from its counterfeit, each being recog- 
nized by itself, without any refining; 3. that it leads to a knowledge of 
the constituents of a metallic body composed of any two metals, without 
separation of one from another, either by melting, or refining, or change 
of form, and that in the shortest time and with the least trouble ; 4. that 
h shows the superiority in weight of one of two metals over the other 
in water, when their weight in air is the same ; and reversely, in air, when 
their weight in water is the same ; and the relations of one metal to an- 
other in volume, dependent on the weight of the two [compared] in the 
two media; 5. that it makes the substance of the thing weighed to 
be known by its weight — differing in this from other balances, for they 
do not distinguish gold from stone, as being the two things weighed ; 
6. that, when one varies the distances of the bowls from the means of 
suspension in a determined ratio, as, for example, in the ratio of impost 
to the value of the article charged therewith, or the ratio of seven to ten, 
which subsists between dirhams and dinars,* surprising things are ascer- 
tained relative to values, without resort to counterpoises — [for instance,] 
essential substance is indicated, and the [mere] similitude of a thing is 
decisively distinguished; and theorems relative to exchange and legal 
tenders and the mint, touching the variation of standard value, and 
certain theorems of curious interest, are made clear ; 7. the gain above 
all others — that it enables one to know what is a genuine precious stone, 

+ Impost, JCwjl y in this passage, denotes the valuation in units of money of a 
unity of provisions of a given sort ; and this unity is the thing charged with the 
impost, JLwtlt . Hence the relation of one to the other must be numerical, like 
that between dirhams and dinars, which are proportioned to one another as 7 to 10. 
This explanation is derived from the second chapter of the eighth lecture of our 
author's work, which is not of sufficient importance to be translated. 
vol. vi. 2 



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10 A r . Khanikoff, 

fi[}& '**&» '&JLA jP} *u» ^cX&h U°j*& J& muLmJI^ *-*Jj£ ^LmmO^ 
L^'u-tf ^ I^Lo sjl ^ fc« jyH 3 1^3 JjJIJj o^ytf X*^ 

vbctft ^ j*>5 *u» jJoJI ^t Lx*J> ^UM 1O49 jU^&kII l$%i* 3 
xl* Ju^Jt ^jJJ ^ ^^^o JjmJ! jj*Jt k** 3 tsr *-o J^o^Jf ^ 

D ^l X^** ^il*!!* q&H c^** **^ ^5^ I* ^^ OjUa* jpl£U 
x&> 3^II Lgjo u^Iju oLji^ ply>t o^Uaj ^lait XaU^uJJ JlSftt 

b^U! UfrJLc Ju*Jud! £1 *LtJ£}t ^Ut 3 XJ& O tj-yo ^yLy* L^JLc^ t^S 

^jjJI 3 A^LaLaj cr li»-X*£d U ^fiit tc\$ q* f^?^ o' ^ J* k^***^' 

such as a hyacinth, or ruby, or emerald, or fine pearl ; for it truly discrim- 
inates between these and their imitations, or similitudes in color, made 
to deceive. 

These views have led us to the consideration of the balance of wisdom, 
and to the composition of this book, with the help of God and His 
fair aid. 

Sec. 2. Theory of the Balance of Wisdom. 

This just balance is founded upon geometrical demonstrations, and 
deduced from physical causes, in two points of view : 1. as it implies 
centres of gravity, which constitute the most elevated and noble depart- 
ment of the exact sciences, namely, the knowledge that the weights of 
heavy bodies vary according to difference of distance from a point in 
common — the foundation of the steelyard ; 2. as it implies a knowledge 
that the weights of heavy bodies vary according to difference in rarity 
or density of the liquids in which the body weighed is immersed — the 
foundation of the balance of wisdom. 

To these two principles the ancients directed attention in a vague way, 
after their manner, which was to bring out things abstruse, and to declare 
dark things, in relation to the great philosophies and the precious scien- 
ces. We have, therefore, seen fit to bring together, on this subject, what- 
ever useful suggestions their works, and the works of later philosophers, 
have afforded us, in connection with those discoveries which our own 
meditation, with the help of God and His aid, has yielded. 



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Book of the Balance of Wisdom. 11 

oK^Lao*} l^JL c ic****. l5^L^ i k feL uo J^ai ^1 JyUi & l^pL* _s 

say* UJ& \ jpsi •» ; ux« x^iUit pun, ja^i l5 ^j ^ 
j^lj? ^ x&ttJt P LoM o^Lao v^!>M ^y^ui ^ut t^ftUAflji 

^s iLo ^ (^OsJt L fr caju ^ gU«*t! ic\0 iiJJUo ^ L^va*© Lujcsd u^jX't 
Jdu J^xse idL ^Ut Ut 3 **I &>lJL OCLfi *sli/s> j^XJI jJUt C-^' 

*S«. 3. Fundamental Principles of the Art of Constructing this 

Balance. 

Every art, we say, has its fundamental principles, upon which it is 
based, and its preliminaries to rest upon, which one who would discuss 
it must not be ignorant of. These fundamental principles and prelimi- 
naries class themselves under three heads: 1. those which rise up [in the 
mind] from early childhood and youth, after one sensation or several 
sensations, spontaneously; which are called first principles, J^d common 
familiar perceptions; 2. demonstrated principles, belonging to other 
departments of knowledge ; 3. those which are obtained by experiment 
and elaborate contrivance. Now as this art which we propose to investi- 
gate involves both geometrical and physical art, uniting as it does the 
consideration of quantity and quality ; and as to each of these two arts 
pertain the fundamental principles mentioned, it has itself, necessarily, 
such fundamental principles; so that one cannot possess a thorough 
knowledge of it, without being well grounded in them. But inasmuch 
as some of the familiar perceptions relative to this art are so perfectly 
evident that it is useless to draw upon them in books, we leave them 
unnoticed ; pursuing a different course in respect to certain first princi- 
ples not perfectly evident, which we shall speak of as there is occasion. 
As for those derivative fundamental principles, obtained by experiment 
and ocular proof, and likewise, as to demonstrated principles belonging 



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12 N. Khanikoff, 

^JLc L$Ii Jo JS>\ jyic £ L^JLt ^j* ^yJi ws5J<A^5 HiA^mi^ io-^iJ 

\jAj2 ajl&t LpjQ ajUXJI ^IjuU 

*Jt ,^0^ ytfJt ^hr JyJfL Ujj ^1 &JLJU v^VJU ^Lt 1 eUJt l#\ 

U^e 0^>^ vK cr* N^ ^ ik*S* &yt* u>j>li Kca&j v-aP3 q* iJt o>^ 
(j^uX-m-wj^I "5ft did & &£sM aOUfc \&jK cX>! *gj£ uX>^j J6 ^UJ .jC 

to other sciences, we shall call them up so far as may be necessary, in 
the way of allusion and passing notice. 

Sect 4. Institution of the Water-balance ; Names of those who have 
discussed it y in the order of their succession ; and Specific Forms of 
Balances used in Water, with their Shapes and Names. 

It is said that the [Greek] philosophers were first led to think of setting 
up this balance, and moved thereto, by the book of Menelaus addressed 
to Domitiaflpn which he says : " King, there was brought one day to 
Hiero King of Sicily a crown of great price, presented to him on the 
part of several provinces, which was strongly made and of solid work- 
manship. Now it occurred to Hiero that this crown was not of pure 
gold, but alloyed with some silver ; so he inquired into the matter of 
the crown, and clearly made out that it was composed of gold and silver 
together. He therefore wished to ascertain the proportion of each metal 
contained in it ; while at the same time he was averse to breaking the 
crown, on account of its solid workmanship. So he questioned the geom- 
etricians and mechanicians on the subject. But no one sufficiently skill- 
ful was found among them, except Archimedes the geometrician, one of 
the courtiers of Hiero. Accordingly, he devised a piece of mechanism 
which, by delicate contrivance, enabled him to inform king Hiero how 
much gold and how much silver was in the crown, while yet it retained 
its form." That was before the time of Alexander. Afterwards, Menelaus 



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Booh of the Balance of Wisdom. 13 

-^SJj ^^lilo *-o Jaj j j\*£m$\ J*** j$> ^ itf-JaJ aJL^. Juuu^> 

i^vXll ^t & &*&& JjJW »Uwj yi*. ^1 vUT £ lP/6 xJU, *** 
j AgwI - .Js+mA} y>i^U^ otjtoit ^>|y>5 v*^ *** t * y *j£ o^ 1 ^' -^ 

**^ CT^ ***'" A * fl * ^ ' JM2 ^\5 ^t^ ^ ^Us* U£> (j&ju ^fi lfc a« j 
x3j i&jul itf£*£\ v3^>-' £* ^5^ **^ Kfl^ fciOtjQi ^^jj^uX+Jt 

^-,jt* *U *j J^«J^ TfcX*^ &5U0 ^^ ^jJ»3 «*$ ^yiJi sjifi^ (^1^ 

[himself] thought about the water-balance, and brought out certain uni- 
versal arithmetical methods to be applied to it ; and there exists a treatise 
by him on the subject It was then four hundred years after Alexander. 
Subsequently, in the days of Mamun, the water-balance was taken into 
consideration by the modern philosophers Sand Bin 'Alt, Yuhanna Bin 
Ynsif and 'Ahmad Bin 'al-Fadhl the surveyor ; and in the days of the 
Samanide dynasty, by Muhammad Bin Zakarlya of Rai, who composed 
a treatise on the subject, which he speaks of in his Book of the Eleven, 
and named this balance " the physical balance ;" and in the days of the 
Dailamite dynasty, by 'Ibn 'al-'Amld and the philosopher 'Ibn-Slna, both 
of whom distinguished [the components of] a compound body scientifically 
and exactly, but composed no work on the subject; and in the days of 
the house of Nasir 'ad-Din, by 'Abu-r-Raihan, who took observations on 
the relations of [different] metallic bodies and precious stones, one to 
another, as indicated by this balance, and carried his deductions so far as 
to distinguish one from another [in a compound], exactly and scientific- 
ally, without melting or refining, Dy arithmetical methods. 

Some one of the philosophers who have been mentioned added to the 
balance a third bowl, connected with one of the two bowls, in order to 
ascertain the measure, in weight, of the rising of one of the two bowls in 
the water : and by that addition somewhat facilitated operations. Still 
later, under the victorious dynasty now reigning — may the Supreme 
God establish it! — the water-balance was taken up by the eminent teacher 
'Abu-Hafs TJmar 'al-Khaiyaml, who verified what was said of it, and 



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14 N. Khanikoff, 

crb^?* cw j A»» ^ c*^** *** ^|)3 **|;' or ^J* ** i)^' i^f ^ 

tJUftjj fly&ywl tOj+fc ^^ifc ojjUJt J^lyO ^>j qL^! Jl jLfclj ^UJj^ 

^1 ^ Uxc, *^>t ^^ ^ \so\mJL xl*^ A' j^ >* *>' * e&*° ^ 
aIII *r> ^1 c5^5 *^ o!i** "^ olr^' ^* <-5>" l^U*l cr ^ 

crfJ^' o ! l^; 1 ^ b** f P ^ *** **LU o!*^ J^5>^ c5* 

«v>^4* J* otjAJLi qj<A^j 'U^ g^-«*H vJiUall q'j-4' *1 Jl^M ek*?**** 
l5^5 <*5j^"-^ v^-^J &\bjJU LfjcX^J oLdjb oU^ v£&3 ^3 i^^ls 

demonstrated the accuracy of observation upon it, and the perfection of 
operation with it — supposing a particular sort of water to be used — 
without having a marked balance. The eminent teacher 'Abu-Hatim 
'al-Muzaffar Bin 'Isma'il of 'Isfazar, a cotemporary of the last named, also 
handled the subject, for some length of time, in the best manner possible, 
giving attention to the mechanism, and applying his mind to the scope 
of the instrument, with an endeavor to facilitate the use of it to those 
who might wish to employ it. He added to it two movable bowls, for 
distinguishing between two substances in composition; and intimated 
the possibility of the specific gravities of metals being [marked] upon its 
beam, for reading and observation, relatively to any particular sort of 
water. But he failed to note the distances of specific gravities from the 
axis, by parts divided off and numbers ; nor did he show any of the 
operations performed with them, except as the shape of the balance 
implied them. It was he, too, who named it " the balance of wisdom." 
He passed away, to meet the mercy of the Supreme God, before perfecting 
it and reducing all his views on the subject to writing. 

Sect 5. Forms and Shapes of the Water-balance. 

Says 'al-Khazini, coming; after all the above named, — Balances used 
in water are of three varieties of shape : 1 . one with two bowls arranged 
in the ordinary way, called " the general simple balance" ; to the beam 
of which are frequently added round-point numbers ; 2. one with three 
bowls for the extreme ends, one of which is suspended below another, and 



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Book of the Balance of Wisdom. 15 

aJLul* ijiaju jJI L^caju otjiill yu o tejuo ^^ °U$w4y .-e qUJLaJ^o 

4^* ji^-4-SI oLilj sUp ^1 &*& Jai (^» JJJ *-oj> &uLJ xlxgg: x^U-'i .Jt 

sLJt ^jLam q^O rb!>> CV>^^ *^ *alilN vj s-AAwU-* yoyos? *U &J^ 

oljUJIj *j^l j<V *«y o ! o^ J-oUlf Last tj^t kx# ^^ 
*V o** t/ 1 * 3 ^ o->** ^ J*"k HvX * y^^ l5*~ a** 1 * o 1 !; 

-1*5 B^Uit j^jJt Juafe ^JU Jt 3 f3L«£! v^T a jJj^ LiJd! Jm aL'! 
^U,Ut ^ »L^Lo ^ ,^u~ e^ll 3I s,»Ut JUKI v3***, H/4jJI KUt 
U*** «JLr ^13 «.!vXX3l s^cLtoj xjLLiLm dJI *bl oV^' r#^ %*^' O^J* 
A3 |pm*JS «.t^it (j^ o^cWaam! JLfcxSG* (jdt a^j aI<Acj ^Aj» r *ajf j^J! ^lit)t 

is the water-bowl ; which is called " the satisfactory balance," or " the 
balance without movable bowl ;" 3. one with five bowls, called " the com- 
prehensive balance," the same as the balance of wisdom ; three of the 
bowls of which are a water-bowl and two movable bowls. The knowl- 
edge of the relations of one metal to another depends upon that perfect- 
ing of the balance, by delicate contrivance, which has been accomplished 
by the united labors of all those who have made a study of it, or pre- 
pared it by fixing upon it [points indicating] the specific gravities of 
metals, relatively to a determined sort of water, similar in density to the 
water of the Jaihun of Ehuwarazm, exclusively of other waters. 

It is also possible, however, for one who is attentive and acute, by 
means of this balance, to observe the specific gravities of precious stones 
and metals [marked] upon it, with any water agreed upon, at any time, 
with the least trouble, at the shortest notice, and with the greatest facility 
of operation ; as I shall set forth in the course of this book, with the help 
of the Supreme God, and the felicity of the imperial power of the most 
magnificent Sultan, the exalted Shah of Shahs, the king of subject 
nations, the chief of the Sultans of the world, the Sultan of God's earth, 
the protector of the religion of God, the guardian of the servants of 
God, the king of the provinces of God, designated as God's Khallf, the 
glory of the course of the world and of religion, the shelter of Islamism 
and of Muslims, the arm of victorious power, the crown of the illustrious 
creed, and the helper of the eminent religion, 'Abu-l-Harith Sanjar Bin 
Malikshah Bin 'Alparslan, Argument for the Faith, Prince of the Be- 
lievers — may God perpetuate his reign, and double his power ! For his 
felicity is the illuminating sun of the world, and his justice its vivifying 
breath. 



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16 N. Khanikoff, 

u*LS!$ mL^UmJI (^» aj ^ibu Jul 2uoi» U JjjiXjt *#g xW-£Ji ^iuuf 

<1Iju *1N J^y*^3 xmmJ^ v^V° fo+5* L 5?>^** m0 3 >SUiJ vV^I ^y^ xSLuLm 
^yj au! *xLumJj JuUaL*; ai^XSj »ji* ^ <Xy^ **>*« ^ aI J-Jaj ^t 

(V *^ oJ^' ^A^ 3 !? o'"**^ o!i**^ *«^$ qv^JI r v-jL»jw5 o^U^t^ 

j+G. Q-t L»£a* JJS^Ij \Jl£»^t &r**J obu L Ji (jmLJU^ olsuUit^ *|j$!t 

I sought assistance from his beams of light irradiating all quarters of 
the world, and was thereby guided to the extent of my power of accom- 
plishment in this work, and composed a book on the balance of wisdom, 
for his high treasury, during the months of the year 515 of the Hijrah* 
of our Elect Prophet Muhammad — may the benedictions of God rest 
upon him and his family, and may he have peace ! 

This book is finished by means of his auspiciousness, and the felicity 
of his high reign, embracing all sovereignties, by virtue of the Supreme 
God's special gifts to him of fortitude and valor — so that he has subdued 
the climes of the East and West — and the excellencies united in him, 
purity of lineage, nobleness of nature, exalted nationality, and lofty 
grandeur, both by inheritance and conquest. So then, may God perpetu- 
ate his reign, who is the chief of the people of the world, the possessor of 
all the distinctions of humanity ! We ask the Supreme God that he 
would lengthen his days, and increase his eminence, his power, his rule, 
and his sway — God is equal to that, and able to bring it to pass. 

Sect 6. Division of the Book. 

I have divided the book into three parts: 1. General and funda- 
mental topics: such as heaviness and lightness; centres of gravity; 
the proportion of the submergence of ships in water; diversity of the 
causes of weight ; mechanism of the balance, and the steelyard ; mode 



* A.D. 1121-22. 



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Book of five Balance of Wisdom. 17 

^5 u£m» ^ ^ <jaju ^yo L^aoju ja*? 3 ojjliH vJiS^' ^ ju J**Jt 

^^KK J^JL v^cJt ^ 3 oyJi vl# o* **» ^ J ^1^(5 l|^LAl q* 
'Ljju eJUS^ ««l** vSj^J J* J.»£*j ,*«&» Lx* 3 d^UUtt, okUJi 

^Ijax^ ( Js , !(l ^iauwJt ^jl^° J^ {J°$ *j>**2 oIt^5 oL^vJukflJi aI^Lm^ 
^juUj^ ^PVj^ v_aJ5 ^1 iu> [ y *j y^y* ^ +4 &»* * & (jJ iauJd lj vjyu 
.Igi 3 t JuJ q* fc-ytoltl oULJ! *j Uyu oLcLmJ! q|ja** oliUj c£*Iaj 
U^-J^ s; A5Ij Lfi jiUail £*a«u 3 ,y|jSH 3 vJb.l*Xib U^**^ 
aJUu Jf 3 o^Uu ^US ylygriB^Ldt SJk>t 3 KJliU ^ JuX&j. ^ 3 

of weighing with it in air and in liquids ; the instrument for measuring 
liquids, in order to ascertain which is the lighter and which the heavier 
of two, without resort to counterpoises; Knowledge of the relations 
between different metals and precious stones, in respect .to [given] 
volume; sayings of ancient and modern philosophers with regard to 
the water-balance, and their intimations on the subject. This part in- 
cludes four lectures of the book in their order. 2. Mechanism of the 
balance of wisdom ; trial of it ; fixing upon it of [the points indicating] 
the specific gravities of metals and precious stones ; adoption of coun- 
terpoises suited to it ; application of it to the verification of metals and 
distinguishing of one from another [in a compound], without melting or 
refining, in a manner applicable to all balances ; recognition of precious 
stones, and distinction of the genuine from their imitations, or simili- 
tudes in color. There are here added chapters on exchange and the 
mint, in connection with the mode of proceeding, in general, as to 
things saleable and legal tenders. This part embraces three lectures. 
3. Novelties and elegant contrivances in the way of balances, such as : 
the balance for weighing dirhams and dinars without resort to counter- 
poises ; the balance for levelling the earth to the plan* of the horizon ; 
the balance known as "the even balance," which weighs from a grain 
to a thousand dirhams, or dinars, by means of three pomegranate- 
counterpoises ; and the hour-balance, which makes known the passing 
hours, whether of the night or of the day, and their fractions in minutes 
and seconds, and the exact correspondence therewith of the ascendant 
star, in degrees and fractions of a degree. This part is in one lecture. 
vol. vi. 3 



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18 N. Khamkoff, 

C)!i*" ^^ (i^9. kS^ ' '^^ j iu^AJ^Jl olrfvXfiil j-5 
- . ..... .. ^ £- 

3 C5^' U»h** oUj.Uf (j»L5* ^ j 

The book is therefore made up of eight lectures. Each lecture in- 
cludes several chapters, and each chapter has several sections, as will be 
explained by the following table of contents, if the Supreme God, who 
is the Lord of Providence, so wills. 

Table of Contents of the Book of the Balance of Wisdom, called 
the Comprehensive Balance, in Eight Lectures. 

Lecture First. 

Fundamental Principles, Geometrical and Physical, en which the Com- 
prehensive Balance is based. In Seven Chapters. 

Chap. 1. Main Theorems relative to Centres of Gravity, according to 
Ibn 'al-Haitham of Basrah and 'Abu-Sahl of KuhistAn, in Nine Sections. 
Chap. 2. Main Theorems, according to Archimedes, in Fonr Sections. 
" 8. Main Theorems, according to Euclid, in Two Sections. 
u 4. Main Theorems, according to Menelaus, in Two Sections. 
a 5. Statement of Divers Theorems relative to Heaviness and 
Lightness, in Three Sections. 

Chap. 6. Theorems relative to the Ship and the Proportion of its 
Submergence, in Four Sections. 

Chap. 7. Instrument of Pappus the Greek for measuring Liquids, in 
Six Sections. 



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Book of the Bah/nee of Wisdom. 



19 



MAI xJUuJ? 

"v_i|^t i u>« > (CjLfcwUt jALJU qU&I X*i«»5 

UbUI g-k* o!i**"^ ^^ 8 b^° l$* 

iu jsiJt^ ^uy 3 UfiJi x*^> ^ 

Oj> C5 3 ' OJ* O* (^ O 1 ** 31 ik^" L5* 

K&SISSI KJUJt 

^UfcUlj A*>jJL L|3l^J 3 X-otXJl oljUil yy«o ^ 



* -**-?^ ^ U* 3 ** ^ 4-* 50 ** *r"»h '^yf^ j*Mi ***j t5* v 

Lecture Second. 

Explanation of Weight and iti Various Causes, according to Thabit; 
Fundamental Principles of Centres of Gravity; and Mechanism of the 
Steelyard, according to 'al-Muxaffar of'Isfazar. In Five Chapters. 

Chap. 1. Quality of Weight, and its Various Causes, according to 
1Mbit Bin Kurrah, in Six Sections. 

Chap. 2. Explanation of Centres of Gravity, in Four Sections. 
u 3. Parallelism of the Beam of the Balance to the Plane of the 
Horizon, in Five Sections. 

Chap. 4. Mechanism of the Steelyard, its Numerical Marks, and Ap- 
plication of it, in Five Sections. 

Chap. 5. Change of the Marked Steelyard from one Weight to an- 
other, in Six Sections. 

Lecture Third. 

Bdations between different Metals and Precious Stones in respect to 
[Given] Volume, according to 'Abu-r-Raih&n Muhammad Bin 'Ahmad 
of Birun. In Five Chapters. 

Chap. 1. Relations of the Fusible Metals and their Weights, proved 
by Observation and Comparison, in Six Sections. 

Chap. 2. Observation of Precious Stones and their Relations to one 
another in respect to [Given] Volume, in Four Sections. 



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20 A r . Khamkoff, 

V *aS UuLi ^5^x31 ^*Jt yjj iotp- ^ l^jj^ 

iuuUI ^UuJf 

^LJl ^ f UUi .,fi*S ^X» «^l JU *U» D t^ J . 

C%ap. 3. Observation of Substances occasionally required, in Two 
Sections. 

Chap, 4. Observation of a Cubic Cubit 1 of Water, Weight of a Vol- 
ume of the Metals one Cubit cube, and Weight of so much Gold as 
would fill the Earth, in Three Sections, 

Chap. 5. Dirhams doubled [successively} for the Squares of the Chess- 
board, Depositing of them in Chests, their Preservation in a Treasury, 
and Statement of the Length of Life in which one might expend them, 
in Two Sections. 

Lecture Fourth. 

Notice of Water-balances mentioned by Ancient and Modern Philosopher*, 
their Shapes, and the Manner of Using them. In Five Chapters. 

Cftap. 1. Balance of Archimedes, which Menelaus tells o£ and Manner 
of Using it, in Four Sections. 

Chap. 2. Balance of Menelaus, and the Ways in which he distin- 
guished between Metals compounded together, in Three Sections. 

Chap. 8. Exposition of what Menelaus the Philosopher says respect- 
ing the Weights of Metals, in Two Sections. 

Chap. 4. Notice of the Physical Balance of Muhammad Bin Zakarlya 
of Rai, in Three Sections. 



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Book of the Balance of Wisdom. 21 

*W ^ ^ 

Chap. 5. "Water-balance in the Form spoken of by the Eminent 
Teacher TTmar 'al-Khaiyaml, Manner of Using it, and its Basis of 
Demonstration, in Four Sections. 

Lecture Fifth. 

Mechanism of the Balance of Wisdom, its Adjustment, Trial of it, and 
its Explanation, In Four Chapters, 

Chap. 1. Mechanism of its Constituent Parts, as indicated by 'al-Mn- 
zaffar Bin 'Isma'il of 'Isfazar, in Four Sections. 

Chap. 2. Adjustment of its Mechanism, and Arrangement of the 
Connection between its Constituent Parts, in Four Sections. 

Chap. 3. Explanation of it, and Express Notice of its Names and the 
Names of its Constituent Parts, in Four Sections. 

Chap. 4. Trial of it, and Statement of what happens or may happen 
to the Weigher in connection therewith, in Six Sections. 

Lecture Sixth. 

Selection of Appropriate Counterpoises; Mode of Operating thereby, in- 
cluding : 1. Discrimination between Mixed Metals, by means of the two 
Movable Bowls, and Distinction of Bach One of two Constituents of 
a Compound, scientifically, with the least labor, 2. Jtpthmetical Deter- 
mination [as to Quantity] of the Two; and Prices at which Precious 
Stones have been rated. In Ten Chapters. 

Chap. 1. Selection of Appropriate Counterpoises, as regards Lightness 
and Heaviness, in One Section. 



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22 A". Khamkoff, 

\ ^J^j tX*yb *aU jP^ilj oUtf /U oL^i **slS J> ^ 

<*£**# jki q* i*2JU q* L^woju ** Ju5^ll j*.1*b *i^^ *Xa*l ^t 

Jucuw &SO^± jdftA*St vJLjOuwt (j^ V-lLwjilj LgJuU JAA4 ^ fl ^9 S 

IOS-J W fA J*W 3 JiW <^> ,3* ^j** 8 V-^ ^ 3 

CAap. 2. Levelling of the Balance of Wisdom, Mode of Weighing 
Things by it, and Application of Numbers to the Conditions of Weight, 
in One Section. 

Chap. 3. Mode of fixing upon the Balance [the points for] the Specific 
Gravities of Metals and Precious Stones, by Observation and the Table, 
in One Section. 

Chap. 4. Knowledge of the Genuineness of Metals, by use of the two 
Movable Bowls, as well as of Precious Stones, whether in the State of 
Nature or partly Natural and partly Colored, and Discrimination of one 
Constituent from another of a Compound, without melting or refining, 
with the least labor and in the shortest time, provided they are com- 
pounded Two and Two, without any thing adverse, in Three Sections. 

Chap. 5. Arithmetical Discrimination between Constituents of Com- 
pounds, through Employment of the Movable Bowl, in the plainest 
way, and by the easiest calculation, and its Basis of Demonstration, in 
Six Sections. 

Chap. 6. Relations between Metals in respect to two Weights : 
Weight in Air and Weight in Water, and their Mutual Relations in 
respect to [Given] Volume, when the two [compared] agree in Weight, 
one with the other, ascertained by Pure Arithmetical Calculation, with- 
out Use of the IfeLance, in Two Sections. 

Chap. 1. Cei^n Singular Theorems, in Two Sections. 
" 8. Knowledge of the Weight of two Metals in Air, when they 
agree in Water-weight, in Two Sections. 

Chap. 9. Certain Singular Theorems, and Knowledge of a Metal by 
its Weight, and the reverse, in Three Sections. 

Chap. 10. Statement of the Values of Precious Stones in Past Times, 
as given by 'Abu-r-Raihan, in One Section. 



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Book of the Balance of Wisdom. 23 

!»4aI^ r***-^ J**"^ XmmJ J^fi fc-h^J; oI^UaoH XLlmI^ -a£ 

vj jJLilXju^ vJyoJf q'j*° (^y^ LS* ^ 

- oL^UaoB J&wt} -si q* ^+A&H **r**5 Oj*aii ^ ^ 

<> vJmgJI J->1-**^ v^ajLcj yy^' ^1%> J^L*** ^ * 



o Lk^0l 



Lecture Seventh. 



Exchange-balance ; Adjustment of it, for any determined Relation as to 
the Weight of Dirhams and Dindrs, by Suitable Counterpoises; 
Knowledge of Exchange, and of the Value of any Metal or Precious 
Stone, without Resort to Counterpoises; Adjustment of it to the Rela- 
tion between Impost and the Article charged therewith, as also to that 
between Price and the Article appraised; and Settlement of Things by 
means of it. In Five Chapters. 

Chap. 1. Statement concerning Relations, and their Necessity in the 
Case of Legal Tenders, in Four Sections. 

Chap. 2. Adjustment of the Exchange-balance, and Levelling of it, 
in Two Sections. 

Chap, 8. Weights of Dirhams and Dinars, estimated by Suitable 
Counterpoises, in Four Sections. 

Chap. 4. Exchange, and Knowledge of Values without Resort to 
Counterpoises, in Tliree Sections. 

Chap. 5. Theorems pertaining to the Mint, and Singular Theorems 
relative to Exchange, in Four Sections. 

Lecture Eighth. In Eight Chapters. 

Chap. 1. Balance for weighing Dirhams and Dinars, without Resort 
to Counterpoises, in Four Sections. 



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24 Ni Khanikoff, 

0liUj V^JLfc t;UO^ 

UJ UjLxJl Q+ Julfi Loj &>j+fi &JUA0 ^ otLiJl olr^ L$* ^ 

1 lP^y-4 UjUI ** JmmJIj sJUUil ^ *U*> ^ c 

jyddt v!*L» oUl*B 

C%op. 2. Earth-balance, Levelling of the Earth's Surface parallel with 
the Plane of the Horizon, and Reduction of the Surfaces of Walls to a 
Vertical Plane, in One Section. 

Chap. 3. Even Balance, and Weighing with it from a Grain to a 
Thousand Dirhams or Dinars, by means of three Pomegranate-counter- 
poises, 2 in Four Sections. 

Chap. 4. Hour-balance, Mechanism- of its Beam, and Arithmetical 
Calculation fputl upon it, in Two Sections. 

Chap. 5. Mechanism of the Reservoir of Water or Sand, and Matters 
therewith connected, in Seven Sections. 

Chap. 6. Numerical Marks and three Pomegranate-counterpoises, in 
Five Sections. 

Chap. 7. Knowledge of Hours and their Fractions, in One Section* 
" 8. Mechanism of the Delicate Balance, and Employment of it 
for Times and their Fractions, in One Section. 

In all, Eight Lectures, Forty-nine Chapters and One Hundred and 
Seventy-one Sections.* 

* Although our author has taken pains to define by synchronisms the periods of 
most of the philosophers whom he refers to in his introduction and table of con- 
tents, I think it proper to add, here, some more exact intimations of the dates 
which concern them. 

The Hiero mentioned must be Hiero ii., who died B. C. 21ft, at the age of Dot 
less than ninety years ; and our author is evidently wrong in placing Archimedes 
before the time of Alexander the Great. It is well known that the great Greek 
geometrician was killed at the taking of Syracuse by Mareellua, B. C. 212. Euclid 
composed his Elements about fifty years after the death of Plato, B. 0. 847. Mene- 
laus was a celebrated mathematician of the time of Trajan, A. D. 08-1 17. But I have 
not been able to find any notice to guide me in identifying ])flm&tiyanus. Pappus 
was probably cotemporary with Theodosius the Great, A. D. 879-895. The philos- 
ophers of the time of Mamun must have lived between A. D. 818 and 888. The 
great geometrician Thabit Bin Kurrah was born in the reign of Mutassim, A. H. 221 
(A. D. 885), at Harran, and died at Baghdad A. H. 288 (A. D. 900). See 'ItoHBbal- 



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Book of the Balance of Wisdom. 26 

Before going farther I must endeavor to discover who our 
author may have been, thus supplying a deficiency occasioned 
by his too great modesty. 3 

Our author continues as follows : 
'ic^^ji tSjcstA U>y\X3 o3Uu *j>t ^JLc J^Xiu Aj^Jiit IlXPj afi^ Jw^ 9 

^i xiuat 

^ v)jS% JUHffl jfys J^U* ^^ KbL>SJ G t Uui>xJi jill^ J^i 

We now enter upon the First Part of the book, relying upon God, and 
imploring benedictions upon His Prophet Muhammad and his family. 
This Part includes four Lectures, which we shall set forth distinctly and 
dearly, if the Supreme God so wills. 

Lectube Fiest. 
Fundamental Principles, Physical and Mathematical. 

We say — God ordering all things by His Providence — that the com- 
prehension of the main theorems relative to centres of gravity, and 

» 

EHtfs Wafayat, ed. De Slane, p. 147. Aa to Muhammad Bin Zakariya of Rai, lie 
is said to have died A. H. 820 (A.D. 982), at a great age. [See Wiistenfeld's Gesch. 
<t Arab. Aerzte u. Naturforscher, p. 41.] Consequently, he was cotemporary with 
ffsir fib Ahmad the Samanide. According to Haji Khalfah, 'Ibn 'al-'Amid died 
A.E360 (A.D. 970), so that he was cotemporary with the Dailamite Rukn 'ad- 
Bnlah. The same authority gives us the date A.H. 428 (A.D. 1036) for the 
forth of 'Ibu-Sina. See Haji Khalfae Lex, ed. Flugel, iv. 496. The Habfb 'as- 
Srar of Khondemir places it in the Ramadhan of A. H. 427, at Hamadan, where I 
nv hit tomb, in ruins, in 1862. 'Abu-r-Raih&n Muhammad, "surnamed 'al-Biruni, 
tocaose originally of the city called Birun, in the valley of the Indus, passed his 
Tooth, and perhaps was born, in Khariam. He was one of the society of savans 
mimed in the capital of Kharizm, at the court of the prince of the country, and of 
itich the celebrated Avicenna f'Ibo-Sina] was a member. Avicenna, so long as 
be fired, kept up relations of friendship with him. When Mahmud undertook his 
expeditions into India, 'al-Biruni attached himself to his fortunes, and passed many 
ttara of his life in India, occupied in making himself master of the Indian sciences ; 
w also endeavored to instruct the Hindus in Arab science, by composing certain 
treatises which were translated into Sanskrit" See Reinaud's learned paper in 
Joora. Asiat for Aug. 1844, 4°" Serie, iv. 123. Mahmud of Ghaznah, as is well 
hwwn, made his first expedition into India A.D. 1000. 'Abu-Hafs 'Uraar 'al- 
Khuyftmi, author of an algebraic treatise lately translated by Woepcke, was born, 
•eroding to a learned notice by this savant in Journ. £siat. for Oct.-Nov. 1854, 
5m Serie, iv. 348, at Nisapur, and died in that same city A. D. 1128. 

Being limited to the resources of ray own library, I am unable to assign more 
definite dates to the other philosophers mentioned in this treatise. 
vol. vi. # 4 



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2ft .•.l.r&Khan&qfi \ 

iy>k\ jLzM &#* J* *ut i » f Uo^« u^jh '*&> d&tt 

_a» «uil*J *\y*2 >H*»Js Kj^»t JjjL* ^JLe (^ iJ^> moU Ou1aa3U 
j^u "i 10^ HjJiw U>li1 IfUtfJ^ «>>5 lij-x* J^U! <*Uj J* old U 

v|>jt ***** j* j-t^Ai Lj* 4y^!> oyj^' <M v*** W* 

1*ju ,J,SI v^ 

^jJb ^bd! y^ £t tjui X^U b>& d^. ^^JJ^ &*£& f^J\ $£\ 3 
lout X|^t ^ //I Kb* il **>a *j* *I ^B j^ W^ 1 o' <y^ 
«yiJS «5J^ K$it y$JL> ^ iC^ £ «^&J! u5JLj ^y^G % j?JL\ ly> j4L 

heaviness and lightness, and the mode of difference in respect thereto, in 
a liquid and in air, and relative to submergence and floating — I mean, 
the comprehension of what is known, in general, respecting heaviness and 
lightness, and the sinking of heavy bodies in water, considered in the 
light of traditions accepted on authority, is of very great utility with, 
reference to the science of the balance of wisdom, and facilitates the 
conception of its ideas. ^Vfter that, on reverting to those theorems, as 
an investigator of the grounds of demonstration on which they rest, 
one lays hold of them by a simple act of thought, without any toiling 
in all directions at once. What is to be said on these main theorems 
occupies seven chapters. 

Chapter First. 

Main Theorems relative to Centres of Gravity, according to 'Abb-Said of 
Kuhistan and 'Ibn 'al-Haitham of Basrah — to aid the Speculator in 
the Science of the Balance of Wisdom to the Conception of its Ideas. 
In Nine Sections, 

OBOXION Fxbst* 

1. Heaviness is the force with which a heavy body is moved towards 
the centre of the world. 2. A heavy body is one which is moved by an 
inherent force, constantly, towards the centre of the world. Suffice it to 
say, I mean that a heavy body is one which has a force moving it towards 
the central point, and constantly in the direction of the centre, without 
being moved by that force in any different direction ; and that the force 
referred to is inherent in the body, not derived from without, nor sepa- 
rated fiom itrr-tbe body not resting at any point out of the centre, and 



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Book of the Balance of Wisdom. 27 

• 

|Jt*J! /y At ^^04 I £1 UbL *U* ^ U J Jul Ljj 
^ lilt ±«xl\ 
jAm^A ±z ^ *i>3 U L|4 <j^ **** iM fUtf>*3 <pt 

JJttB jujUx* JLCtfi t %L&stt L^ X^Uslt ^UtU ^Jl »L*uJ^ 

^^11 AAk^at lf*»jJ, Vl5Uv\^ ^s^mJ JjJt ^ ^gjSJt 

s-4-^u l^ft *xTy> D li iU^ f Lo*l £ iV-JB ^mo- d^O' 13^ ^ 

J 4j*& ^ 5sS g^-J v^t ^ A u^> u^ta L^b^ 

6 s&tttft Uk^ JX&R l^lAx* r ^i! I^Uju a U**>. v^ ^ 

G U^> ^a Ut 3 eJUJJ £j»\ yjti *** u^^M ( ***£l *^>>> 

^t ^ ^^JJt ls Jj&ft Uki? sytfl i L 3 Uju f^i l^Ux* 

being constantly moved by that force, so long as it is not impeded, until 
it reaches the centre of the world* 

Saonoir Ssoown. 
1. Of heavy bodies differing in force some have a greater force, which 
are dense bodies. 2. Some of them have a less force, which are rare 
bodies. 3. Any body whatever, exceeding in density, has more force. 
4. Any body whatever, exceeding in rarity, has less force. 5. Bodies 
alike in force are those, of like density or rarity, of which the correspond- 
ing dimensions are similar, their shapes being alike as to gravity. Such 
we call bodies of like force. 6. Bodies differing in force are those which 
are not such. These we call bodies differing in force. 

Ssonoir T&nu>. 
1. When a heavy body moves in liquids, its motion therein is propor- 
tioned to their degrees of liquidness ; so that its motion is most rapid in 
thai which is most liquid. 2, When two bodies alike in volume, similar 
in shape, but differing in density, move in a liquid, the motion therein of 
the denser body is the more rapid. 8. When two bodies alike in volume, 
and alike in force, but differing in shape, move in a liquid, that which 
has a smaller -superficies touched <iy the^fiquid- nfovel therein -more 



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28 K KJuznikoff, 

M^LmuCa oliLwwQ AJ^LmJC* XJUli £ ^>?^J' Lp' (5^' JUjL*Ju9 lgJBj> 

^ .^^C, jJUJt ^T^ l3 ^51*35 ^T^ J* ^^^C, J^fiS ^^^ J^ JJJ\ 

L^LwJwQ bLyO Jb\jtl\ jfjA ^il &jlf> fj»> £0 ^U>l J^ O^b *^!5 

j^^l 1+** *H$ J^ (M«*fij jJUt yy cr gj^ <i^' c^ 3 ^ 5 ' J^ cv^* 

^^OJi -jlxwJt J(j y^LJt giiiJt <*5J*3 JOc Jjttlt ^^Ijw ^^wju 

gJa*JI dJv3 uXJLfc JittSI ^v>Ljw ^ac (J^v^Sj ****&» >SL*J1 j5^r jc m ^3 *kfl*J 

rapidly. 4. When two bodies alike in force, but differing in volume, 
move in a liquid, the motion therein of the larger is the slower.* 

Section Foueth. 

1. Heavy bodies may be alike in gravity, although differing in force, 
and differing in shape. 2. Bodies alike in gravity are those which, when 
they move in a liquid from some single point, move alike — I mean, pass 
over equal spaces in equal times, 3. Bodies differing in gravity are those 
which, when they move as just described, move differently ; and that 
which has the most gravity is the most rapid in motion. 4. Bodies alike 
in force, volume, shape, and distance from the centre of the world, are 
like bodies. 5. Any heavy body at the centre of the world has the 
world's centre in the middle of it ; and all parts of the body incline, with 
all its sides, equally, towards the centre of the world ; and livery plane 
projected from the centre of the world divides the body into two parts 
which balance each other in gravity, with reference to that plane. 
6. Every plane which cuts the body, without passing through the centre 
of the world, divides it into two parts which do not balance each other 



* The MS. reads gw»t , but it is evident that our author would say *l 



Hut. 



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Booh of the Balance of Wisdom. 29 

JU!I jfjA ^Lc vJuIaJU *JU ^yJt KkifcJt ^13 JuJS' /*J^=> J(j gLJt 
,.^♦^1 «*JJJ J£tft /^ ^^^ *Jlfc LtfU a K lot 

IM ^ o J0JL» UP X*jyU xkai JU* JJBtt ^>UxJt uU^t, 5» 
^fc LgiSS |^y ^Loj *Jl53 jfj* xLiUi! ii5Ja yu J^ p**3* il L*>t 

^ U*>t 3t ^ !«3JJt U$> u^j^ gL~ uu* Jifctit yA a ^U45 
tfJJ £»> ^axXj S \ gL*Jt w^O^ jdS3 j*> a ya J*J8 ^.^o* 
J!^«3 eJUJt gL*Jt «5Uo ^ £*^t JJiS /j* OJ ^ ^*JL\ 

^ a fe U^S^y* ^tfj. ^ j^tl u*i> JU* XbLiArf JUBt £! ^t 
gb~ JU«s XbU** JlSSt it ^t *% j^li-t j^Jt «£L> JUft aJoLkXxi 

with reference to that plane. 1. That point in any heavy hody which 
coincides with the centre of the world, when the body is at rest at that 
centre, is called the centre of gravity of that body. 

Section Firm. 

1. Two bodies balancing each other in gravity, with reference to a 
determined point, are such that, when they are joined together by any 
heavy body of which that point is the centre of gravity, their two [sepa- 
rate] centres of gravity are on the two sides of that point, on a right * 
line terminating in that point — provided the position of that body [by 
which they are joined] is not varied ; and that point becomes the centre 
of gravity of the aggregate of the bodies. 2. Two bodies balancing 
each other in gravity, witn reference to a determined plane, are such that, 
when they are joined together by any heavy body, their common centre 
of gravity is on that plane — provided the position of that body [by which 
they are joined] is not varied ; and the centre of gravity common to all 
three bodies is on that plane. 3. Gravities balancing each other rela- 
tively to any one gravity, which secures a common centre to the aggre- 
gate, are alike. 4. When addition is made to gravities balancing each 
other relatively to that centre, and the common centre of the two is 
not varied, all three gravities are in equilibrium with reference to that 
centre. 6. When addition is made to gravities balancing each other, 
with reference to a determined plane, of gravities which are themselves 
in equilibrium with reference to that plane, all the gravities balance with 



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80 N. Khanikoff, 



U*j> J^Liu v)uJG jm^> J(^ gLJt aJj>LjuU jUSUI! ^ gn^J JJB jSyo 

sjo^ Kbfc ^ I^JUtfl jilyi j>Lju! ^1 JbCa» %UaJ\ ^Jz>*1\ £ 
KftUril, JJKJt *xblittsj xLiiJI 41? Al wUo'JfL JJttlt KbUu £ x^Ux. 

w ^/^^L?m ^ *ju ^un/^ A! ^u J*tt <**> X "55 

reference to that plane. 6. When subtraction is made from gravities 
balancing each other, of gravities which are themselves in equilibrium, 




and so the centre of gravity of the aggregate is not varied, the remaining 
gravities balance each other. 1. Any heavy body in equilibrium with 
any heavy body does not balance a part of the latter with the whole of 
its own gravity, nor with more than its own gravity, so long as the posi- 
tion of neither of the two is varied. 8. Bodiesaalike in force, alike in 
bulk, similar in shape, whose centres of gravity are equally distant from 
a single point, balance each other in gravity with reference to that point ; 
and balance each other in gravity with reference to an even plane passing 
through that point; and such bodies are alike in position with reference 
to that plane. 9. The sum of the gravities of any two heavy bodies is 
greater than the gravity of each one of them. 10. Heavy bodies alike 
in distance from the centre of the world are such that lines drawn out 
from the centre of the world to their centres of gravity are equal. 

Section Sixth. 
1 . A heavy body moving towards the centre of the world does not 
deviate from the centre; and when it reaches that point its motion 
ceases.* 2. When its motion ceases, all its parts incline equally towards 

* This proves that the theory of momenta was. entirely unknown to the Arabs 
of the twelfth century; excepting for the cam of thel«v«r. v -'- v •*■'•'—"-- - 



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Book of the Balance of Wisdom. 81 



l«&u ^ JUS f Lc>l /It ^1 ^ 5Ji 3 ff y l ^^ 1 *ju fj.\ 

ff LJt >*M 
^Wtt JUa UfcX»4 £&, JtfL$U Jd*|j LgJUi (jOLJii £&***»• vK J^ 

Jy-u^^! ^^6 * ^ Jail J* litt jJ> ^^yu ^ ^ LfrLu ^ 

JjUi xiis JJgJt £ *} ^L** ^^ y a ls 3LSS U*o* J^Liu JuJS p^o* 

Jv>Ui ^ iliB f^J-\ J^bu * uli *u JJtft ^io- *!»/,<• Jot ^ 

au* JJBt U^> lit 

the centre. 3. When its motion ceases, the position of its centre of 
gravity is not varied. 4. When several heavy bodies move towards the 
centre, and nothing interferes with them, they meet at the centre ; and 
the position of their common centre of gravity is not varied. 5. Every 
heavy body has its centre of gravity. 6. Any heavy body is divided by 
any even plane projected from its centre of gravity into two parts bal- 
ancing each other in gravity. 7. When such a plane divides a Dody into 
two parts balancing each other, in gravity, the centre of gravity of the 
body is on that plane. 8. Its centre of gravity is a single point. 

Section Seventh. 
1. The aggregate of any two heavy bodies, joined together with care as 
to the placing of one with reference to the other, has a centre of gravity 
which is a single point 2. A heavy body which joins together any two 
heavy bodies has its centre of gravity on the right line connecting their 
two centres of gravity ; so that the centre of gravity of all three todies 
fc on that line. 3. Any heavy body which balances a heavy body is 
balanced by the gravity of any other body like to either in gravity, when 
there is no change of the centres of gravity. 4. One of any two bodies 
which balance each other being taken away, and a heavier body being 
placed at its centre of gravity, the latter does not balance the second 
body; ft Waooertrajy ; 4J^o$r of 91016 gravity than ifcat has. 



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32 K. Khantkoffy 

j& jdii J^=>j* qIs *W^' *jU**rf -jia^J! ^LwJU ^***^> J^ vV^ 

LA\*|y> J* UpUij^ qI^Ux* U^U^^ tt^fiit vj j^L*** g>b*Jt 
U*X>t ^ jUm^^I Jitf &\ L0J«>i Jitf iU«*S Q lj K*ld Li^j ^L* 

p»k^Ji {£)]}** Cte**^ &» j»Ju> w*jLu* q» (2^JULaX^ ^j^h^ *\j[y* 

fj* Sr 5 ***"* <* & »"* 3*& \+&* &*&* <***£' "*!i*}* s/* ^ 

£jyf ^i^-iYib j^i\ ^\ l*x>i Ja^ai L5 *~s ju^jtf ^i ^i 

JU*.tfy>tf JJtf ^1 Uax>1 JJtt jU*J a fe uOU^o (jOLfiS c^-m^ ^ 
U^ju *o»i, JjCJ ^jJdt e£UH tails*! /\j* *uU ^t Ja^JS ^^m-3 

SsCtTOlf EtQBTtf. 

1. The centre of gravity of any body having like planes and similar 
parts is the centre of the body — I mean, the point at which its diame- 
ters intersect 2. Of any two bodies of parallel planes, alike in force, 
and alike in altitude — their common altitude being at right angles 
with their bases — the relation of the gravity of one to the gravity of 
the other is as the relation of the bull* of one to the bulk of the other. 
3. Any body of parallel planes, which is cut by a plane parallel with two 
of its opposite planes, is thereby divided into two bodies of parallel 
planes ; and the two have [separate] centres of gravity, which are con- 
nected by a right line between them ; and the body as a whole has a 
centre of gravity, which is also on this line. So that the relation of the 
gravities of the two bodies, one to the other, is as the relation of the two 
portions of the line [connecting their separate centres of gravity and 
divided at the common centre], one to the other, inversely. 4. Of any 
two bodies joined together, the relation of the gravity of one to the 
gravity of the other is as the relation of the two portions of the line on 
which are the three centres of gravity — namely, those pertaining to the 
two taken separately, and that pertaining to the aggregate of the two 
bodies — one to the other, inversely. 

Section Ninth. 

1. Of any two bodies balancing each other in gravity, with reference 
to a determined point, the relation of the gravity of one to the gravity of 
the other is as the relation of the two portions of the line which passes 



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Book of the Balance of Wisdom. 33 

^ i&Si» <aU* ^ # ^^vXJt Jail (J? **o xm*uT y>fl Jjfi Jl UWot 

C U>1« e^JLsa <^> j^ JsiSi yMB ^1 Lpjo*! u^ib ^/^ 

J* *X« Utf juli jTi! aJb j^vXJt p-J&Ut Jait JLt tl&jfjA Uuj 

JUfll jfelj* JuUwQ v^ 

through that point, and also passes through their two centres of gravity, 
one to the other. 2. Of any two heavy bodies balancing a single heavy 
body relatively to a single point, the one nearer to that point has more 
gravity than that which is farther from it 3. Any heavy body bal- 
ancing another heavy body relatively to a certain point, and afterwards 
moved in the direction towards that other body, while its centre of grav- 
ity is still on the same right line with the [common] centre, has more 
gravity the farther it is from that point. 4. Of any two heavy bodies 
aKke in volume, force, and shape, but differing in distance from the centre 
of the world, that which is farther off has more gravity. 

End of the theorems relative to centres of gravity. 

The second chapter is entitled Theorems of Archimedes with 
respect to Weight and Lightness. I shall not give a translation 
of it, since it contains notning which is not known. Our author 
commences by quoting from the Greek geometrician, though 
without specifying the work from which he derives the quota- 
tion, to the effect that different bodies, solid and liquid, are dis- 
tinguished by their respective weights; then he proceeds to 
enunciate, without demonstration, the principle of Archimedes, 
that the form of a liquid in equilibrium is spherical ; that a 
floating body will sink into the water until it shall have displaced 
a volume of water equal in weight to its own entire weight ; 
and finally, that if a body lighter than a liquid be plunged into 
that liquid, it will rise from it with a force proportionate to the 
difference between the weight of the submerged body and that 
of an equal volume of the liquid. 



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84 & Khanxkaff, 

The title of the third chapter is Theorems of Euclid respecting 
Weight and Lightness, and respecting the Measuring of Bodies 
by one another. It contains sundry geometrical definitions re- 
specting volumes, the enunciation or the well-known equation of 
dynamics expressing the relation between the velocity of motion, 

the space traversed, and the time, t;= j , and that of the princi- 
ple that gravity acts upon a body in the direct ratio of its mass. 

The fourth chapter has for its title Theorems of Menelaus 
respecting Weight and Lightness. It contains only a few well- 
known developments of the principles of Archimedes applied to 
solid and hollow bodies (v^mo* (»~> and uj^» f»~>), and I shall 
do no more than cite from it some of the technical terms made 
use of. The water into which a solid is plunged is called 
^uL\ *uJt , " like water," if the water displaced by the immersed 
body be of the same weight with that body ; and the latter is 
designated JJU3 rr^» Ulike *xdj" ff of lcss weight, the 
body is called \-~~\j\ f Jil , " sinking body;" if of greater weight, 
it is styled ^^C\ T £\ , " floating body." 

The fifth chapter contains a recapitulation of the principles of 
centres of gravity, and is here given entire, with a translation : 

iyee XilS ^ J^X&j^ a LJL5 *>Urf JoUm* £ 
J^!Ju*aJI 

Js#>> ^ ^jsofcj l^cou XJ^Ijm y^jL&i *S )Um£l^i\ f>}f>X\ a 5 J**' 

A&L> J sJtfSl J> J>\ ^ \6\ iujOaJt f \j£\ vjbL^U ±*£y /J* 

kXP, ±&\ y^j ubttS ^ Utrt\ cr 1^> Z*>\ s*u otfTfl j& 

Chapter Fifth. 

Theorems recapitulated for the sake of Explanation. In Three Sections. 

Section First. 

Difference in the Weights of Heavy Bodies, at the same Distance from (he Centre 

qf the World. 

1. I say that elementary bodies — differing in this from the celestial 
spheres — are not without interference, one with another [as to motion], 
in the two directions of the centre and of the circumference of the 
world, [as appears] when they are transferred from a denser to a rarer 
air, or the reverse. 2. When a heavy body, of whatever substance, is 



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Book of the Balance of Wisdom. 85 



l^u* vJkSttl J^l J>\ V^ > Oj^ A ^ ^^ erf/**" 

jfcLu. k*& L*** Ufraajo ^bu £»U J d^" til Jy^t-jMu^JI J^W 
*Y> yub aitt, *^5 ^^ jus ^ ^^J J*fi£J! ^iJI f/ > *UJ 

D > >» rfnii a tr> > w 3 iys» ^ ***« o!** a w>wj 

a?)^' r^" ls>^ c**M *W o> ^°^ &>* **** o^ *^t '*** & 

transferred from a rarer to a denser air, it becomes lighter in weight ; 
from a denser to a rarer air, it becomes heavier. This is the case uni- 
versally, with all heavy bodies. 3. When one fixes upon, two heavy 
bodies, if they are of one and £he same substance, the larger of the two 
in bulk is the weightier of them. 4. When they are of two different 
substances, and agree in weight, and are afterwards transferred to a 
denser air, both become lighter ; only that the deficient one, that is, the 
smaller of the two in bulk, is the weightier of them, and the other is 
the lighter. 5. If the two are transferred to a rarer air, both become 
heavier ; only that the deficient one, that is, the smaller of the two in 
bulk, is the lighter of them in weight, and the other is the heavier. 

Ssonoir Ssoomx 

1. When a heavy body moves in a liquid, one interferes with the 
other ; and therefore water interferes with the body of any thing heavy 
which is plunged into it> and impairs its force and its gravity, in pro- 
portion to its body. So that gravity is lightened in water, in propor- 
tion to the weight of the water which is equal [in volume] to the body 
having that gravity ; and the gravity of the body is so mucn diminished. 
As often as the body moving [in the water] is increased in bulk, the 
interference becomes greater. This interference, in the case of the bal- 
ance of wisdom, is called the rising up [of the beam]. 2. When a body 
is weighed in air, and afterwards m the water-bowl, the beam of the bal- 
ance rises, in proportion to the weight of the water which is equal in 
volume to the body weighed ; and therefore, when the counterpoises are 
proportionally lessened, the beam is brought to an equilibrium, parallel 



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36 Ni Khamhff, 

\Juii\ &la*» H\j]yA J* *>y*}\ &»*& *jA*l oL^U^I or <jaft \6\ kX$J 5 
*Y>. ^kXiu v^^p. vJjA* ^1 *a£M vJ (.^ j&~\ !j^ gtjJJ Uw? 

^>^t L(Ub cr JJ«t »W^ A k*V** & *W* k^ vM^ii P tj*M 

lot a5&L» J^ JrfuiJtf Uhfall ^ iH vi^Ifii I3t 3 gUJJ <*U3 £ 

v.^>t v^Jtf vjtfrf *5j0 ^\ v^Jiu 

*iU ^UJI j£ay» q* u^a5? Juui a ^t pin* J*S3 fj > ^ ^1 
^ iV* 3 * oK °^ 5 a tf UK *ili JUrf «Oum u&*M v^-rfl fcX3^ vjdX*** 

^VouuiJ su^T Jtfli ^l JttR ***** oj£=* && v-**t o tf y£ 
^ JLbfj* o* gr^ i5*B ^*^ 5 <> tf 3 *^>» o^ ^ 

t^ Uo jaS»1 U^mmIj i^rf X*L*U j^i ijoji\ gkm j$2* o* K i,i hr 

with the plane of the horizon. 3. The cause of the differing force of the 
motion of bodies, in air and in water, is their difference of shape. 4. Yet, 
when a body lies at rest in the water-bowl, the beam rises according to 
the measure of the volume of the body, not according to its shape. 
5. The rapidity of the motion of the beam is in proportion to the force 
of the body, not to its volume. 6. The air interferes with heavy bodies ; 
and they are essentially and really heavier than they are found to be in 
that medium. 7. When moved to a rarer air, they are heavier ; and* 
on the contrary, when moved to a denser air, they are lighter. 

Seotiok Thud. 

1. The weight of any heavy body, of known weight at a particular 
-distance from the centre of the world, varies according to the variation 
of its distance therefrom ; so that, as often as it is removed from the 
centre, it becomes heavier, and when brought nearer to it, is lighter. 
On this account, the relation of gravity to gravity is as the relation of 
distance to distance from the centre. 2. Any gravity inclines towards 
the centre of the world ; and the place where the stone having that 
gravity falls, upon the surface of the earth, is its station ; and the stone,' 
together with its station, is on a straight line drawn from the centre of 
the world to the station mentioned. 3. Of any two like figures, stand- 
ing on one of the great circles of the surface of the earth, the distance 



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Booh of {he Balance of Wisdom. 



37 




JXi ^L> ^yua^v^Jt ^UUu ^ ti!> Lf L^ *j\x*1% jJUit jS^-s U^ 

OJ ia -i vk& ,Jl*it /^ a* W t> 

C?/" U>^ cjl^S 0"? l* a J*»** *UJt «>5 

between the apexes is greater than that between the bases, because the 
two are on two straight lines drawn from the centre of the world, making 
the two legs of a triangle, of which the apex is the centre of the world, 
and the base [includes] the two apexes. When the stations of the two 
figures are connected [by a right line], we get the shape of two simi- 
lar triangles, the longer of which as to legs is the broader as to base. 
4. The place of incidence of a perpendicular line from the centre 
of the world, falling upon any even plane parallel with the horizon, is 
the middle of that plane, and the part of it which is nearest to the 
centre of the world* Thus, 4 let the plane be a b y the centre h, and the 
perpendicular line upon ab from the centre he — that is the shortest 
one between the centre and the plane. 5. Let any liquid be poured 
upon the plane a 6, and let its gathering-point be h, within the spherical 
surface a hb 9 [formed by attraction] from the centre A, then, in case the 
volume of the liquid exceeds that limit, it overflows at the sides of ab. 
Una is so only because any heavy body, liquid or not, inclines from 
above downwards, and stops on reaching the centre of the world ; for 
which reason the surface of water jayiot flat, but, on the contrary, con- 
vex, of a spherical shape. On this account, one who is on the sea, with 
a lighthouse in the distance, first descries its summit, and afterwards 
makes out to discover, little by little, what is below the summit, all of 
which was before, as a matter of course, concealed ; for, excepting the 
convexity of the earth, there is nothing to hide every other part but the 
summit, in the case supposed. 0. Let any sphere be formed by gravita- 



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88 N. KhxmUcoff, 

t> &Lft3 ^JLc v_A£j ^» y>lXjj >«lX2Xjj _j>i>J5 «—i\ gia** ^Lfc 

Jftl <y U o^jfi ^ l+*Hr* v^ x *&i\ uJj ^ £u\J\ 8^C5i gin- 

tion over the plane ab — after being so formed, and oscillating to and 
fro, it stops at a point d, contrary to the opinion of those who think that 
it is accumulated and oscillates perpetually. 7. Of liquids in receptacles, 
they contain a greater volume when nearer to the centre of the world, 
and when farther from it contain a less volume. Thus, let abh be [the 
bulge of water in] a receptacle, at the greater distance from A [the cen- 
tre], and within the spherical surface of the water, a h 6, over tie top of 
the receptacle [by attraction] from the centre of the world, let the 
liquid in the hollow of the receptacle be contained, and let a section of 
the surface of the sphere — which you perceive to be not a plane — be 
ahb by a z 5, and let the right line between these two be zh\ and, on 
the other hand, let there be a receptacle at the less distance tz, in case 
we fix upon t as the centre of the world, and let the new section of the 
surface of the sphere be arfft, over the top of the receptacle, [by a* 6], 
and let the right line between these two be zd. So then, what is in the 
receptacle increases by the excess of zd [over «A], namely, the interval 
between two spherical surfaces at different distances from the centre of 
the world — which is what we wished to state. 



I shall not stop to point out certain inaccuracies in the fore- 
going theorems of centres of gravity, since each reader will 
readQy discover them for hirdfelf ; but I will observe, in gen- 
eral, that the vagueness of the ideas of the Arab physicists 
respecting force, mass, and weight, a vagueness which is the 
principal cause of these inaccuracies, seems to have troubled 
them very little, for our author is no where at the pains to estab- 
lish a distinction between those three ideas. But the ideas of 



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Booh of the Balance of Wisdom. 89 

the Arab philosophers with regard to gravitation are, in my opin- 
ion, much more remarkable ; I will not call it universal gravita- 
tion, for oar author expressly exempts the heavenly bodies from 
the influence of this force (see Chapter Fifth, Sect. First, 1.), * but 
terrestrial gravitation. That this great law of nature did not 
present itself to their minds in the form of a mutual attraction of 
all existing bodies, as Newton enunciated it five centuries later, 
is quite natural, for at the time when the principles exhibited by 
our author were brought forward, the earth was still regarded as 
fixed immovably in tne centre of the universe, and even the 
centrifugal force had not yet been discovered. But what is 
more astonishing is the feet that, having inherited from the 
Greeks the doctrine that all bodies are attracted toward the cen- 
tre of the earth, and that this attraction acts in the direct ratio 
of the mass, having moreover not failed to perceive that attrac- 
tion is a function of the distance of the bodies attracted from 
the centre of attraction, and having even been aware that, if the 
centre of the earth were surrounded by concentric spheres, all 
bodies of equal mass placed upon those spherical surfaces would 
press equally upon the same surfaces, and differently upon each 
sphere — that, in spite of all this, they held that weight was 
directly as the mass and the distance from the centre of the 
earth, without even suspecting, so far as it appears, that this 
attraction might be mutual between the bodjr attracting and the 
bodies attracted, and that the law as enunciated by them was 
inconsistent with the principle which they admitted, that the 
containing surface of a liquid in equilibrium is a spherical sur- 
fece. Many geometricians of the first rank, such as Laplace, 
Ivory, Poisson, and others, have endeavored to establish the con- 
sequences of an attraction which should act directly as the dis- 
tance from the centre of attraction ; thus Poisson says :* "Among 
the different laws of attraction, there is one which is not that of 
nature, but which possesses a remarkable property. This law is 
that of a mutual action in the direct ratio or the distance, and 
the property referred to is this, that the result of the action of 
all the points of a body upon any one point is independent of 
the form and constitution of that body, whether homogeneous 
or heterogeneous, and is the same as if its whole mass was con- 
centrated in its centre of gravity." Farther on, he shows that 
under the influence of this law the containing surfaces of a re- 
volving liquid are ellipsoid or (with sufficient velocity) hyperbo- 
krid ; tne latter form being possible as a permanent figure only 
•rben the liquid is contained within a vessel It is thus seen that 
none of the immediate effects of an action in the direct ratio of the 
iistanoe were of such a character as to set the Arab philosophers 
m their guard against the consequences of their law of terres- 



* Trait* de MScanique, 2*« Edition, iL 550*553. 



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40 2f. Khanikoff, 

trial gravitation, for they had not the means of arriving at these 
conclusions. On the other hand, the principle of Archimedes, 
and the suspicion which they had of the different density of the 
atmosphere at different heights, taught them that the farther a 
body was removed from the earth's surface, and consequently 
from its centre, the less of its weight it would lose from the 
effect of the medium, that is to say, the heavier it would become ; 
they did not, therefore, hesitate to admit the direct ratio of the 
distance. It is evident that the Arabs admitted the heaviness of 
the air, and even that thev had, so to speak, discovered the 
means of estimating it, for they say that a given body loses less 
of its weight in a rarer than in a denser atmosphere ; but in 
all probability they never made application of this means to 
ascertain the weight of a volume of air at different altitudes. 
Finally, neither the Greeks nor the Arabs, so far as appears, 
were in possession of any positive demonstration of the princi- 
ple according to which a liquid in equilibrium takes the form of 
a sphere, but they admitted it as an evident principle, founded 
on the spherical form of the surfaces of great sheets of water. 
Upon the whole, it seems to me allowable to believe that the 
Arabs had one great advantage over the ancients with respect to 
the study of nature ; this, namely, that they were to a much less 
degree than their predecessors in civilization bent upon fitting 
the facts observed into artificial systems, constructed in advance, 
and that they were vastly more solicitous about the fact itself than 
about the place which it should occupy in their theory of nature. 

I shall make no extracts from the sixth chapter, which pre- 
sents nothing at all worthy of note, but shall pass directly on 
to the seventh chapter. 

It reads as follows, in the original and translated : 

iS ^J\ j*i^ f*£&U ** 0s*% a*£|* J^iil £ oUjUI ^Liu XaJUo £ 

Chapter Seventh. 

Mechanism of the Instrument for measuring Liquids, as to Heaviness or 
Lightness, and Application of it, according to the Philosopher Pappus 
the Greek. 

It is evident, from the theorems already stated, and from what is to 
be presented respecting the relations between the gravities of bodies, 
that the relation of any volume of a heavy body to any volume of an- 
other heavy body, in direct ratio, when the two weigh alike in air, is as 

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Booh of the Balance of Wisdom. 41 

ij: uw«j U {$&& *Jt L^jy^iu -j&jJmjJ* x*.iJLw,4 &*\a£t *xp ^L>o 

^^f £ Lptj^t o^l tit ^4^ q^PIj (JSWU ^Jt L^yflJU obj-bjit ^rt *-> 
^JLoj ,jrJt ^L*£$t ^p tuV> £**j} ;aJ 4 v ^* c U* 30 ** lf^> y*$} l-^" 
l^jcXiij ju& jfvXJ& Q^sy^ oL^Uao JLjOaJ ^a£ q* (jnLJI qL\jJ &5=UaJ 

a*y&, JuJt ^ J ^a*a3 ^tOJU Xit^t JXA L§&£ jdl XWt *XP <y* t 
iu^ys^ AA«jtfM ^a£ *3^° u"L^ cr ^ ^ J 5 ' ^' O****' U^y* >** 
^ivAj q U^ » L*a,«j>» (jv-djJLtJt q* qUsXcIS L^5j &*A (M^J U v^fc>t rf**^ 

JjC& jJ££ ^mJU mm -b~> ^XS u^U^j Jj>tJJt (jO'uXelfcJt ^jiX&A fla** 

the relation of gravity to gravity, inversely, [when the two are weighed] 
in water. The force of this fundamental principle, once conceded, leads 
to the construction of an instrument which shows us the exact relations 
in weight of all liquids, one to another, with the least labor, provided 
their bodies are of the same volume, definitely determining the light- 
ness of one relatively to another ; and which is very useful in respect to 
things concerning the health of the human body ; and all this without 
resort to counterpoises or balance. We shall, therefore, speak of the 
construction of this instrument, the marking of lines upon it, and the 
development of a rule for the putting upon it of arithmetical calculation 
and letters [expressive of numbers] ; also, of the application of the 
instrument, and of its basis of demonstration. This will occupy six 
sections. 

Section Fast. 

Construction of the Instrument 

The length of this instrument, which is cylindrical in shape, measures 
half a hand-cubit ; and the breadth is equal to that of two fingers, or 
less. It is made of brass, is hollow, not solid, and the lighter particles 
of brass are carefully turned off by the lathe. It has two bases, at its 
two ends, resembling two light drum-skins, each fitted to the end, care- 
rally, with the most exact workmanship ; and on the inner plane of one 
of the two bases is a piece of tin, carefully fitted to that plane by the 
lathe, shaped like a tunnel, the base of which is the drum-skin itself. 
vox* vi. 6 



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42 



-K Kkmikojfi 



Form of Pappus the Greek's Instrument for Measuring Liquids. 6 

t 



between ( \*Xg\jo JwVjJ! ^a** 
sb and hr. | Ascending Line of Numbers. 

between ( ^ ^^ ! °^ 
wm&ndkr.) Descending Bound-point 7 
( Numbers of the Instrument. 



Pto" 



akh. 



above 
akh. 

below 
akh. 



along the 
line of de- 
scending 
numbers. 



on the 
cone at the 
lower end 
of the in- 
strument. 



above the 
cone. 



Equator of Equilibrium. 
Lighter Side. 
Heavier Side. 

iujiUJt alx&U rfj»» 

Different Part-numbers 

sought for, determining the 

Weight of the Liquid. 

Cone made of Tin. 
■^ vAjlJI ^J! vXiuil ju«J 

The Relation of Distance to 
Distance, successively, from 
the Base, is, by an Inverse 
Ratio, as the Round-point 
Numbers of the Second 
Distance to the Round-point 
Numbers of the First Dis- 
tance. 



U" 



-I 



Uf- 



=, mi 



f 



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Book of the Balance of Wisdom. 48 

ij> '^^Oj ^ *S& v^juto} tot ,J&> aJUsu vJOJt u&O JuvXftlSj ^jjjJ^o 
i^sjL^- ^1 J**? ^ LuqaJU UL3 julc i&slS *\J\ $\ \jcy>- 

L frJlc JaJa^wJt _s 

^ ^L l* Jst ^ L|A*l5 (jMX-^tjJu j^u ^i Jtail top ^ Jfii! ^5 

xa-jA* B^uummq JSc>^ Xtaftj ^Ics KASaj^ t^ JJU Jpj ^o ^ J^Ia3» 
vt" Ja3* ^^*J J I^JU Jitffl vuL?- "V**** 1*5 JUtffl JL>1 vuL> 

V^ JO L*k^" L5^ ^^ fL**^' -k& ^ ji*\$ J^?^ f 1 ***' V^ 
»U: jq J»- f ^J i Y* ^q Jix> q^ j»U*ii »yi4U ^j>^*3 J^^Lo fM&i} 

The instrument being thus made, when put into liquid in a reservoir or 
vessel, it stands upon it in an erect position, and does not incline any way. 

Swnow Ssoonp. 

Marking of Lines upon the Instrument 

You draw, in the first place, lengthwise, along the whole cylinder, a 
line sab, forming its side; and let the upper part of this line remain 
above f water] in the vessel, namely, a small piece measuring a sixth of 
the height of the cylinder, or less, as, making a part of the line [sab] 
contiguous to the line of one of the bases, * 'a. You also draw other lines 
parallel with the line a b, namely, the lines rh, wm, ht, extending to 
the limit [Va] mentioned. Moreover, you bisect the line a b at k, and 
lay off each of the lines tr, md, It, equal to it a, and over the points 
Jfc, t, d, I, you place a bent ruler, fitted to the bulge of the cylinder, and 
draw a circular line over those points ; and, in like manner [after laying 
jff the lines jl, tn, khd], you draw, over the points ajnkh, a circle 
ajnkh, which you call the equator of equilibrium. That part of the 
instrument above the equatorial line is the side of lighter gravities 
[than that of water], and that part below the same is the side of heavier 
gravities. 

Afterwards, you divide the line ab into ten parts, for number-letters, 
and over the points of the several parts you make stripe-like arcs, rest- 
ing upon the lines nr and ab ; and you divide the distance between each 
two parts into ten parts, on the line nr, so that the line nr is divided 



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44 N. Khanikotf, 

AJ^LmX* \j\jU6 Lyw£ - Ja Ja> (J^ IgAAj JutOAd AJjLamU LoLmSJ f*wj> 

L** ^1 c> Ia^JI £ wOtfCxJj ^^XeUJI viyUJ iu^ yC^ oU£ft 
Jam Ifcfwi^ T^5V3 ^ cXac q* k^X*** ^ v-*$j^ g-b v-J l^ 3 ^ ^^ 

xWI ^ ^LfiJI p|^>I j^ 3J> 0> sLsii v^-^> s^^ ^ 

slo itffl &ilS UU>5 aJt XfljAV l*^ 3 ? jL~t 3 ? J^> hLo 3 ? JlSU 
(j*LfiJt s\j=A *&& vi^OcS. uu^J bc^t bli iu^>L> *U! vL***^ *■*** 

i jut ^o^jui jut y^ »utus> > aJ? b^a& j^a* sfti j rfu Uj^ *** 

Jc>G lib XfcH ^ v^UJt COuJt ^Llm, q* pj> pp. jU» Ik^l toli 

- .Ui s£*«&>j tuX^f v^aJ! B^ix: JuJlfi fjw&£ OOwJtli ^Ll^ ^ *j|L U^13 

into a hundred like parts. Then yon connect the hundred points of 
that line with the line tj, by small arcs at even distances one from an- 
other, which are consequently parallel with the circles of the two bases ; 
and you are to write on the surfaces [divided off] between the two lines 
a b and tj the [appropriate] number-letters, beginning at b and pro- 
ceeding towards a, which make what we call the line of even number. 

Section Third. 

Arithmetical Development of a Ride for the Proportioned Part-numbers 
[indicating Specific Qravitie*\ and Putting of them upon the Instrument, 

You are now to understand how to find all numbers indicating the 
weights of liquids. In the first place, we fix, in imagination, upon any 
vessel whatever, as, for example, the daurak, capable of containing [a 
weight of] water equal to a hundred mithkals, or a hundred dirhams, 
or istars, or any thing else, at our pleasure ; and we put down, for the 
height of the instrument [to the water-line], one hundred numbers, cor- 
responding to the quantity of water assumed. Then, when we wish to 
make up a table, putting into it the proportioned part-numbers, we mul- 
tiply 100 by 100, producing 10,000, which we keep in mind, it being 
the sum to be constantly divided ; and ifj then, we wish to obtain the 
proportional for each part of the line of numbers marked upon the 
instrument, we take [the number of] that part, from the line of num- 
bers, and divide 10,000 constantly by it, and the quotient of the divis- 
ion is set down, opposite to [the number of] that part, in the table of 
part-numbers and fractions of part-numbers. But that portion of the 



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Book of Hie Balance of Wisdom. 45 

& a tf L> \*)y»£=>> rfp* & J3J4 A *£ <AA3 ty '**»& v* 
^ *UH crM&SjM H^5 U*»s>jp sUit G ^ cOmJl jam 

Xjy3jll^j5 SU5! Oj^^ l* OJuJl Ja-w q^5 f lyuM^H Jai> ^^ ^5^ 

& ^iy* Lo Jl xBl *x* £ S U^. ^* ^ X it o Wrf U«§ sXxJI 

*^ J* LT^* ^Ip"' °l-^ ^ '*% D>^' ^5 vX: ?" jj^*° »X0>j LgJU 
t« tfl3 iM Qf £ ^l *N\ C*\* jam *\y>\ ^ j^> J^ 5 L ^y lib 

O 1$^ 15* O* *x5Utt H^LuJlL «by\> Ui^ U^Lo Ua5 Jttfui) LptjAcj 
Jai> ^j^i L^U j3j Ui \^ v V^^?~ CT* ^ ^hj^ £*>* l5^^ 

line of numbers below 100 is the basis of calculation for liquid heavier 
than water. [So that, for liquids lighter than water, we must have 
numbers above 100 to calculate upon.] 

The basis of demonstration upon which this calculation rests will be 
stated hereafter ; 'Abu-r-Raihan alludes to it in his treatise. 

So much of the instrument as is above the equator [of equilibrium], 
and so much of the line of numbers as is above 100, pertains to liquid 
lighter than water, such as oil and the like. [In our table] we have 
contented ourselves with lines of numbers from 50 to 110, inasmuch as 
this instrument does not require to have upon it [for the calculation of 
specific gravities] numbers either greater than the one or less than the 
other of the two. 

The rule drawn out in a tabular form follows presently. 

When we wish to mark the proportioned part-numbers upon the 
instrument, we set the units of the part-numbers on the line hr, and 
their fifths and tenths on the line wm, in such manner that the [pro- 
portioned] number-letter of each of the parts of the line of numbers on 
the instrument, from 1 to 120, shall be just what the table makes it ; 
and with a bent ruler, in the mode spoken o£ you make lines of con- 
section between hr and wm from 110 to 50. We begin with placing 
the number-letters [derived from the table] on the side of 6, and proceed 
towards a ; but those of them which come above the line of equilibrium 
constitute the measure for light liquid, and those below that line are 
the standard for heavy liquid — both being relative to the gravity of 
»ater. 



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46 







i£ Khanikoff, 










Jjj3Ufl v l 


JM> JjO^?. 






Julio 


*& 


JOuJ! ^ 


vJbLdv^ 


AfA 






*& 


vJ 


Jo 


ua 


cs 3 


*J 


j* 


i* 


&* 


Id 


JaS 


v* 


e* 


e* 


*J 


Wfd 


g* 


e* 


JaJo 


j* 


/ 


f° 


• J 5 


*1 


lo 


^ 


1/ 


Jwa 


.J 5 


«i' 


g* 5 


xc 


^v. 


JU0 


*S 


c 


ii3 


a* 


_b 


y° 


Jo 




J>» 


g* 


» 


J* 


C* 


Ju 


g* 


V** 


UJ 


tr 


V s * 


U 


r 5 


U 


1 


Jp*a 


Li 


■ u 


v*** 


e 




vJ> 


v3 


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JuS 


j^— 


1 


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J340 


s 


J* 8 


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Booh of the Balance of ' -Wisdom. 



47. 



Table of Calculation by. the Bule* 



i*e of Number*. 


Part*. 


Sixtieth*. 


Line of Number*. 


Part*. 


Sixtieth 


110 


90 


54 


80 


125 




109 


91 


45 


79 


127 


35 


108 


92 


35 


78 


128 


12 


107 


93 


27 


77 


129 


58 


106 


94 


21 


76 


181 


85 


105 


95 


14 


75 


133 


20 


104 


96 


9 


74 


185 


8 


103 


97 


5 


73 


187 




102 


98 


2 


72 


188 


54 


101 


99 


1 


71 


140 


51 


100 


100 




70 


142 


51 


99 


101 


1 


69 


144 


56 


98 


102 


2 


68 


147 


3 


97 


103 


6 


67 


149 


15 


96 


104 


10 


66 


151 


30 


95 


105 


15 


65 


158 


51 


94 


106 


23 


64 


156 


15 


98 


107 


31 


63 


158 


44 


92 


108 


42 


62 


161 


17 


91 


109 


54 


61 


163 


56 


90 


111 


7 


60 


166 


40 


89 


112 


21 


59 


169 


80 


88 


113 


38 


58 


172 


25 


87 


114 


57 


57 


175 


26 


86 


116 


17 


56 


178 


34 


85 


117 


89 


55 


181 


49 


84 


119 


3 


54 


185 


11 


83 


120 


29 


53 


188 


40 


82 


121 


57 


52 


192 


18 


81 


123 


28 


51 
50 


196 
200 


5 



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48 N. Khanikoff, 

^^OJ! Sj^JUaL^ c3 A**£Jt »lijX^ j^XJJ LpU> J! ;U*4 ^j^J * g^3 

^LyJI ^ H^rl! u£p£U3 q! j^i o* v^^ My*** k^b *ft^ *-*** 

(joK^t u^UdjjS vj Ju^j \ Ul» iuypJt <j)J3 J J^OUj ^ ^ *UW LIU* 

j^> j^mJL U^j^ *>Wt 3 t ^LaajJ! Jj^3 Lte La J^ v^aaj i^^. *ju 
*LM ^Ja*w (^y^«! t<3l5 Ij^Jj-* l{y^*M» iui^iauwbL! j?y£j i^iXJi * ( mo St ^y^ 
OJb -gj J^U v>a^L o>L*JI 0°r^ '^3 *^' vtf*£ OkAft .ttyuw^ Jai> £* 
*»■ *J! »Ul! jj.L* <j«Lib L?^ ^ uy!jiJS ^t q^>^ <*-^u ^ 

Section Fourth. 

Specification of the Proportion in Weight of the Piece of Tin. 

It is necessary that the piece of tin which has been mentioned, the 
tunnel-like thing upon the base 6 m, on the inner plane of that base, 
should be of such proportion [in weight] that the liquid-balance, when 
put into water, stands even upon it, and sinks, without any agitation, 
either of the liquid or of the balance, until the equator of equilibrium 
is reached, upon which one's determined weight of the liquid is marked, 
as, for instance, the 100 for water in our diagram. In order to deter- 
mine this proportion, experiment is resorted to ; for the tin is either too 
heavy or too light, until the motion of the instrument is arrested at the 
line spoken of; and you carefully reduce the deficit, or the excess, [in the 
weight of the tin,] by the lathe, until the cylinder, being of the allotted 
size, is evenly balanced. When the surface of the water is even with 
the equator [of equilibrium], the instrument is finished. Bo much for 
the determination of the proportion of the piece of tin, adapted to water 
from some known stream of a city or valley, such as the Jaihun or the 
Euphrates, or others — that being taken as the standard for all waters, 
as to lightness or heaviness ; and we may change from one water to an- 
other by varying the weight of the piece of tin, and making observations 
thereupon. Let this, then, be kept in mind. 

Section Fdth. 

Knowledge of tlie Application of the Instrument. 

This instrument is such that, when cast into a liquid not having con- 
sistence, it sinks therein without hindrance ; and if you hold it up erect, 



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Book of the Balance of Wisdom. • 49 

iJi J* w^Lktl itfi^4t *1^>M ^ ^LsK *»js4 q, jy^l ^L 

en o**N \Jy^ *^* a' 4^ ^?^ LpvXXfi JbA#^ *a* v^L 3 t 
; tJiu 2uj £ jdl «U! &\ i^AA Ji>y^ Mud\ JJU U*, fe^Lyt «*JU 
U-o LM ti& ^ ^U! JJ8S A! Js>J ^JuJt UT L^ g^ ^}J\ *Ut 
&&\ J^ *Li fbiM \JUj1 \ li^ Ugp . ^ Ug.9h.il j^ai *J! ^I jJb *U 

UJLt ju£&t i.> slit ^t aU*J oij^J! yJu JJtft ^ JJtf^t v^ 
of^-ufc£Jl ^XXfij^ K<vJiIt q* -3- Ui v^aJI *tyk* ljul *jl* *UI ^y&Lo 

not inclined, it shows the weight of that liquid by some proportioned 
part-number, among the different part-numbers looked for, marked upon 
the submerged line, provided that number is even with the level surface 
of Hie liquid, so that the line of submergence is upon it, or nearly so. 
The number found for the liquid in question you keep in mind, and you 
say that so much of that liquid, with its mark according to the number 
kept in mind, as would fill the imagined daurak, is proportioned to 100, 
the weight of an equivalent volume of water, as the number kept in 
wind is to the specific gravity of water. 

We proceed m the same manner when we compare the water of an- 
other country with that of the stream fixed upon ; and thus it is made 
to appear which of the two is the rarer and lighter in weight If a 
surface of water coincides with any line of less number than kh, that 
water is rarer than water of the stream fixed upon ; and if it exceeds 
kk, that is, comes on the side of greater gravity, it is heavier, by the 
measure of the round points [counted to the surface], relatively to 100. 

Should we be unable to distinguish the number of the round points, 
the line of even numbers is in plain sight By (the number which 
marks] the point of coincidence on this line with the water we divide, 
constantly, 10,000 ; and the quotient of the division is the number of 
the round points looked for. This is what we wished to state. 
VOL. vi. 7 



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60 



N. Khamkoff, 



a 


J 


n 


z 


o 


A 

« 


t 


rf 


I 


{JO 


'a 


8 


i 


m 



^Jto-^OwsuxU Jjfcit s^t *i>W' £ L*!s >>o 

o**>!*» O* 8 ^ 5- O*^^ CJ* ta * iSU O*^ 
JjSlj So^kJl Ja-y»u J* ^^ SU^a^t J^XoU! 



O' 



s 



g £ Jm> Su-J a bli xLasit i^bJt j> it JUuiii 

Ljji ^jju^ ctfjg i£JI *M^ Xj>-t>jJt ***-Jj |»o 

Ka.^ J> H*>>t, XJji»jJt vaotf tot j^c» Xitjla*^ 

Xif^ta# X^L-Jl Li^S ^*Jt XjjbJI o J0 3( ^ Jac- ^t ^ Ja3- 

Ssotion Sixth. 

2?(ww of Demonstration to the Foregoing Statement 9 

Let the cylinder nm be made, and set upon some liquid into which it 
drops down, even and erect, until the line td is reached ; and, on the 
other hand, [let it be pat] into some liquid of great gravity, so that it 
descends [only] until the line J as is reached. Accordingly, each of the 
two lines td and J as — two lines circling round the cylinder, parallel 
with each other, and parallel with the two bases of the cylinder — rests 
upon the level surface of the liquid. I say, then, that the relation of 
the line 'aj to the margin y t is as the relation of the weight of the light 
liquid to the weight of the heavy liquid. For, the relation of the line 
'aj to the line jt is as the relation of the cylinder 'am to the cylinder 
tm ; and so much of the light liquid, in which 'am is held up and im- 
mersed, as is equal in bulk to that cylinder, bears the identical relation 
to so much of the same liquid as is equal in bulk to the cylinder f m, 
which the line J aj bears to the line j t. But [that volume of 1 the 
liquid mentioned which is equivalent to the cylinder tm, equals, in 
weight, [a volume of] the liquid having more gravitv equivalent to the 
cylinder ^am; because these two cylinders [of liquid] are each of equal 
gravity with the whole cylinder nm — as Archimedes has already ex- 
plained in the first lecture of his book on the sustaining of one thing by 



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Book of the Balance of Wisdom. 



61 



^ JJBt £ Jti\ ft *J) J b~? X4UI! Su^jU l^ ^ iyjl— $ f o 

K\Jo»X x» 3 UM Ju^t j, ^ «*XM^ oL^H 
J li^t U «U3 5 *JLSSH "ol^Jl cr tf*«4 f£ 
XWt 8 jL ye ^t £> J LiU \*» yw OS I 5 eJLi 

Ja£- ^Jt\ Jj«u. (_£*>• w**^ ^.^ j** ly**" - * ' Jl *> 

J^^^kWkK^jB eUicr Ob^a35 fe g 

.£ ^i ****** 150 Ja» *tj:>t jkXSc Lj^ji t ^_9 Ja»- 

vy**^ ^dt *4>kfJl qjs *^"*' iVi* ^ ^5 *-»V J 3 ^" (_?^ v' -k"*" 

another — and the relation of y aj to jt is exactly as the relation which 
the weight of the light liquid, equivalent in volume to the cylinder 'aw, 
bears to the weight of the heavy liquid, equivalent in volume to the 
cylinder f am; which is what we wished to explain. 

This having been made clear, we go back to the figure of the instru- 
ment, and say that, if the cylinder st is put into any liquid, in an even 
position, not inclined, and it sinks until the line akh is reached, the 
weight of a daurak of that liquid is according to the measure of the 




part-numbers at the line akh ; and so, when it sinks, in a liquid of more 

"_" "; of that I' _' m 
fling to the measure of the part-numbers at the line fy. For, the 



i — 1 — - _ , 
the line fy is reached, the weight < 



; liquid is 



gravity, [only] until 
according to tne mei 

relation of the line a b to the line bf agreeably to the preceding expla- 
nation, is the relation of the weight of the [lighter] liquid in which the 
cylinder sinks to akh to the weight of the [heavier] liquid in which it 
sinks to/y, inversely. But the relation of the weight of the liquid in 
which the balance sinks to the line fy [to the weight of the lighter 
liquid], is the same as the relation of the number upon the line fy to 
the number upon the line akh\ and the number upon the line/y is the 



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52 Ni Khanikoff, 

/8 ^! xilsU vi^fj ^jM lt*^ ol*>LU ltW* vt r» u?*> o' ^ ^ 

looked for weight of a daurak of that liquid in which the liquid-balance 
sinks to/y ; while the number marked upon the line akh is the weight 
of a daurak with the determined 100 [mithkals, dirhams, istars, or the 
like] of that liquid in which the liquid-balance sinks to akh ; which is 
what we wished to explain. 

The chapter on Pappus the Greek's instrument for measuring liquids is 
ended ; and herewith ends the first lecture. 

The substance of this demonstration, which our author states 
in a somewhat intricate manner, may be presented as follows. 

A floating body always displaces a volume of liquid equal in 
weight to the entire weight of the body itself. The liquid acts 
upward with a force equivalent to this weight, and, the body 
acting in a contrary direction with the same force, equilibrium 
is maintained. If afterwards the same body is plunged into a 
liquid less dense than the former, the part of it which is sub- 
merged will be greater than when it was immersed in the denser 
liquid, because the volume of the rarer liquid required in order 
to weigh as much as the floating body will be greater. The 
absolute weight of these two bodies being the same, their spe- 
cific gravities will be in the inverse ratio of their volumes ; that 
is, g:g'::v , :v) g and g 1 beirlg the specific gravities of the two 
bodies, and v and v' their volumes. The most interesting circum- 
stance connected with the statement of these principles is that the 
author professes to have derived it from the first chapter of a work 
of Archimedes, which he describes as Lsmu Ujcjo *L&K J*s> £ 
" on the sustaining of one thing by another," and which is proba- 
bly the same with his treatise negi tS* tidati hyKnapivwy or negt t©k 

I have copied the figure of the areometer of Pappus as given 
by our author, with the corrections required by nis description 
of it. One may easily perceive that the instrument is nearly- 
identical with the volumeter of Gay-Lussac, and that it was pro- 
vided with two scales, the one with its numbers increasing up- 
wards, to indicate the volume submerged in liquids of different 
density, the other with its numbers increasing downwards, to 
show the specific gravities corresponding to those submerged 
volumes. The table called "table of calculation by the rule" 
merely repeats the same thing. Let us take, for example, the 
line of the first scale marked 88. We find in the table that the 



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Book of the Balance of Wisdom. 53 

line of the second scale corresponding to this division is 113 and 
38 sixtieths ; or, expressing the latter fraction by decimals, 88 
corresponds to 113.63383+. Now it is clear that the specific 
gravity of water, taken as the unit, will be to the specific gravity 
of a liquid into which the areometer sinks to 88, as 88 to 100 ; 
the specific gravity of the latter will accordingly be 1.1363636+, 
or, if multiplied by 100, 118.63686+, which differs from the 
figure adopted by our author by 0.00303, or a fifth of a sixtieth, 
a fraction of wnich his table makes no account. So also 93, 
according to our author, corresponds to 107.51666 +, and ac- 
cording to us, to 107.526881+ , the difference of which is 
0.010215 +, or six tenths of a sixtieth; and so on. From this 
it appears, also, that by adopting the method of sixtieths our 
author gained the advantage of being able to make the figures 
in his table fewer, without affecting thereby the thousandths of 
his specific gravities. In order to understand why he supposes 
that the limits 50 and 110 are more than sufficient for all pos- 
sible cases, I would remark that, as we shall see farther on, the 
Arabs at this period were acquainted with the specific gravities 
of only seventeen liquids^ besides water, which they took for 
their unit, and mercury, which they classed among the metals, 
and not among the liquids. In this series, the heaviest liquid 
was, in their opinion, honey, of which the specific gravity, being 
1.406, fell between 71 and 72 of the first scale of the areometer; 
the highest was oil of sesame, having a specific gravity of 0.915, 
which corresponded on the areometer to the interval between 
108 and 109 ; while the table gives in addition specific gravities 
from 2 to 0.902. 

The table of contents will have already excited the suspicion 
that the second lecture of the work, treating of the steelyard and 
its use, would be found to contain only elementary matters, of 
no interest. In truth, our author exhibits in the first chapter 
the opinions of TMbit Bin Kurrah respecting the influence of 
different mediums upon the weight of bodies transferred to them ; 
in the second chapter he reverts to the theory of centres of 
gravity, and demonstrates that the principle of the lever applies 
equally to two or three balls thrown at the same moment into 
the bottom of a spherical vase ; 10 the third chapter contains a 
recapitulation of what had been already demonstrated respecting 
the parallelism with the plane of the horizon of the beam of a 
balance when loaded with equal weights ; chapters four and five, 
finally, exhibit the theory, construction, and use of the steelyard. 

The third lecture is beyond question the richest of all in re- 
sults, which it is moreover the easier to exhibit, inasmuch as our 
author has taken the pains to collect them into a limited number 
of tables. The first chapter, according tp the table of contents, 
should treat of the relations of the fusible metals, as shown by 



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fet N. KhcmVcoff, 

.observatiou.aad comparison. In my manuscript of the Book of 
the Balance of Wisaom there remain only a few leaves of it, 
tne contents of which cause the loss of the rest to be greatly 
lamented. Notwithstanding the multiplied errors of the copyist, 
oroiasions, and inaccuracies of every kind, which prevent me 
from giving the text itself it is possible to perceive what is being 
treated of, and I shall cite here and there fragments of intelligi- 
ble phrases, which have guided me in the following exposition. 
The first of the remaining leaves, after a few unintelligible words, 
which evidently belong to a phrase commenced upon the pre- 
ceding leaf, contains a figure representing an instrument devised 
by 'Abu-r-RaihSn for the determination of specific gravities. I 

Form of the Conical Instrument of 'Abu-r-Raiian. 

a l^Sic Neck of the Instrument. 

6 &&Jt Perforation. 

c uUt H^yo J^ MjA&f Tube in the Form 

of a Water-pipe. 
' d 1$j$j* Handle of the Instrument. 
€ XWt *i Mouth of the Instrument. 
/ XaiCft */&>* Place of the Bowl [of the 

Balance]. 

give an exact copy of it, * * with all the explanations which ac- 
company it A mere inspection of it will suffice to show that • 
we have here to do with an instrument made for determining the 
volumes of different heavy bodies immersed in the water which 
fills a part of the cavity, by means of the weight of the water 
displaced by these bodies, which is ascertained by conducting it 
through a lateral tube into the bowl of a balance. The descrip- 
tion given by our author of the use of this instrument, which he 
calls qI^N &*} *jj$j&1\ xWl, "the conical instrument of 'Abu-r- 
Baih&n," confirms this idea ; but he adds that the instrument 
is very difficult to manage, since very often the water remains 
suspended in the lateral tube, dropping from it little by little into 
the scale of the balance : *j &dl£ XSyfc *\l\ q* Kite* i^-H *j>*ftW • 
Capillary action was accordingly known to the Arabs; and 
our author asserts that 'Abu-r-Kaihan had ascertained that, if 
the lateral tube had a circular flexure given it, was made 
shorter than a semicircle, and was pierced with holes, the 
water would flow readily through it, no more remaining in the 
tube than just enough to moisten its inner surface: u£J<A^3 




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Book of the Balance of Wisdom. 66 

L 5 ;i! ^3all JJLII q« g}-** 3 ^- 'Abu-r-Kaih&n understood veTy well 
that the size of the neck of this instrument affected the sensi- 
bility of its indications, and he says that he would have made it 
narrower than the little finger, but for the difficulty of removing 
through a smaller tube the bodies immersed in the water, but 
that this size was small enough to mark a sensible variation of 
the level of the water if a single grain of millet were immersed 
in it: U x-LaJL l^o gUJ j$b *^lo$ ^ jaJ& a**^ v£*-^. «ULi> iMj 

In the third section of this chapter, our author gives the re- 
sults of his experiments with the instrument of 'Abu-r-RaiMn 
to ascertain the specific gravity of the various metals. 1. Gold. 
He says that he purified this metal by melting it five times ; after 
which it melted with difficulty, solidified rapidly, and left hardly 

any trace upon the touchstone : (j*-*£> *>l»- m3 jJCj^oL v-aAXJ! cm^ 
j£*£3 (iL^L J^j *>>*->• £/*"!$ *&^ y»* ^5*^ vAj* ; after that, he 
made ten trials, to obtain the weight of the volume of water dis- 

njd by different weights of the gold, and he found, for a hun- 
mithkfils of gold, weights varying from 5 raithkfils, 1 diinik, 
and 1 tassfij, to 5 m., 2 d. : as mean weight, he adopts 5 m., 1 d., 
2 1 2. Mercury. Our author begins by saying that this is not, 
properly speaking, a metal, but that it is known to be the mother 
of the metals, as sulphur is their father. He had purified mer- 
cury by passing it several times through many folds of linen 
cloth, and had found the weight of a volume of water equal to a 
volume of a hundred mithkfils of mercury to be from 7 m., 1 d. f 
1 i t., to 7 m., 2 d., 2$ t. ; of which the mean, according to him, is 
7 m., 2 d., 1 1 3. Lead. The weights found for a volume of water 
equal to that of a hundred mithk&ls of this metal were from 8 m., 
4 d., It, to 9m. ; of which the mean, according to our author, is 
8 m., 5 d. 4. Silver. This metal was purified in the same man- 
ner as the gold, and the weights of the corresponding volume of 
water were from 9 m., 2 d., 2 1, to 9 m., 4 d., 2 t. ; the mean 
adopted by our author being 9 m., 4 d., 1 1 5. Bronze, an amal- 
gam of copper and tin ; the proportion of the two metals is not 
given : the mean weight which he adopts is 11 m., 2 d. 6. Cop- 
per. Least weight, 11 m., 4 d., 1 1. ; mean, 11 m., 8 d., It. 7. A 
metal of which the copyist has omitted to give the name : the 
weights found for it vary from 11 m., 2d., to 11 m., 4 d., 8 (., 
their mean being 11 m., 4 d. ; this value identifies it with the 
metal given as brass in the later tables. From these same tables 
we are able also, by reversing our author's processes, to discover 



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56 



IT. Khanikoff, 



the mean weights adopted by him for the two metals iron and 
tin, respecting which no notices are derivable from this part of 
the manuscript; and for the sake of clearness, and of uniformity 
of treatment with the other classes of substances given later, we 
present annexed, in a tabular form, the water-equivalents of the 
metals, or the weights of a volume of water equal to that of a 
certain fixed weight of each metal respectively. 



Metals. 



Weights of a volume of water equal to that of a hundred 
mithk&U of each metal. 

Reduced to Tassujs. 
136 
177 
212 
233 
272 
277 
280 
310 



MhhkAk 


Daniks. 


Tassujs. 


5 


1" 


2 


7 


2 


1 


8 


5 




9 


4 


1 


11 


2 




11 


3 


1 


11 


4 




12 


5 


• 2 


13 


4 





Gold, 

Mercury, 

Lead, 

Silver, 

Bronze, 

Copper, 

Brass, 

Iron, 

Tin. 

For the reductiou of these weights to the form of an expres- 
sion for the specific gravity, and for a comparison of the specific 
gravities thus obtained with those accepted by modern physic- 
ists, the reader is referred to the general comparative table, to 
be given farther on, at the conclusion of our presentation of this 
part of our author's work. l * 

q\J\ JudJt 
UgJLu Ji&il v-^uwJ ^ 

sit f&J\ £ KjjUwU f\j>>i\ *\P uu>! j) o^jl !Jt u53JJ 3 l&u w 

i+»+tf a> ^j lg^ SCcaAit r/ > y$ &\ v T *fc>J! j*^ a ^ V +m JS *+oai\ 9 

Section Fourth. [Lect. 8, Chap. 1.] 

Relations of Gravity between the two [Metal*]. 

When the volumes of the two agree, and because all water-equivalents 
are related by gravities, the two [water-equivalents of the two metals 
compared] are related to each other by inverse ratio of gravity. There- 
fore, in case one desires to ascertain the weight of one of these bodies 
equivalent in volume to a hundred mithkals of gold — for example, silver 
— the relation, in weighty which the water-equivalent of gold bears to 
the water-equivalent of silver is not as the relation of the weight of the 



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Book of the Balance of Wisdom. 57 

**& **J rr*" ^ *' ****' ^ o>* ^ jf^' ^ f^b **•-• ai> ^ 
*% Mr? 4 oW sr 5 ^ «W» »** »j3« (> -^-» W l^> ^-^ 



rf.. 



lv.1 

ifn 

IMa 

nir 
i.ip 

I. A. 

1v6 
1PP 



D ^Ax^ xL^ *t*j^ v_aJt 
yU USl 3 hUj vjJt 

qjJLMm^ JUm*J>^ ttLiUftJ 






USI 



ktf ! 



O' 



C5^^ 



Ltf 



,L3t 



o 



LSt 






0-. 






a>t> 



0^% 



o*x£ 



body of gold to the weight of the body of silver, but is as the relation 
of the weight of the body of silver to the weight of the body of gold, 
by an inverse ratio. So then, if the weight of the gold is multiplied 
by the weight of its water-equivalent, and the product is divided by the 
weight of the water-equivalent of silver, or of any other body of which 
we wish to ascertain the weight [of an equivalent volume, the desired 
result is attained]. But we have designated a hundred mithkals as the 
weight of the gold, so that the product of the multiplication of that 
into its water-equivalent is an invariable quantity, namely, 525 mithk&ls. 
That number, then, must be kept in mina, in order to the results which 
we aim to obtain, until the division of it by these [several] water- 
equivalents brings out, as quotients, the weights of [equal volumes of] 
the bodies having the [several] water-equivalents. We have done ac- 
cordingly, and have placed the results in the following table. 
VOL. vi. 8 



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58 



K Khanikoff, 



Table of Retult* from Water-equivalent* of Bodies. 



Metals. 

I 
Gold. 

Mercury. 
Lead. 
Silver. 
Bronze. 

Copper. 

Brass. 
Iron. 

Tin. 



Weights of Bodies equal fa 



Volume. 
Mfthkals. Dinifcs. 



100 
71 

59 

54 

46 

45 
45 
40 

38 





1 

2 



2 

3 

8 



T*mty. 


I 

2 

2 





3 



Weights reduced to Taseujs. 

Two thousand four hundred. 
One thousand seven hun- 
i dred and nioe. 
Oue thousand four hun- 
dred and twenty-six. 
One thousand two hun- 
dred and ninety-eight. 
One thousand one hun- 
dred and twelve. 
One thousand and ninety- 
! two. 

One thousand and eighty 
I Nine hundred and seven- 
i tv-five. 
Nine hundred and twen 
ty-two. 



Tssnajs in 
Numerals. 



2400 
1709 

1426 

1298 

1112 

1092 

1080 

975 

922 



The fifth section is entirely wanting in my manuscript As 
regards the sixth section, which is the last one in this chapter, it 
contains a recapitulation of all that had been before stated re- 
specting the specific gravities of bodies, and may be summed up 
in the jamiliar enunciation that the specific gravity of a body 
is the ratio between its absolute weight and the weight of the 
volume of water which it displaoes. 

From this point onward, the condition of the manuscript per- 
mits me to resume the citation and translation of longer extracts. 

Chjjtkr Second. 

Observation of Preeiou* Stone*. In eeveraj Section*, 

u Men prise these metals," says 'Abu-r-Raihan, " only because* under 
the action of fire, they admit of being made into conveniences for then, 
s«ch as vessels more durable than others, instruments of agriculture^ 
weapons of war, and other things which no one can dispense with who 
is set to possess himself of the good things of hie, and ia desir- 



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Booh of the Balance of Wisdom. :69 

i^a^II ^Lib vJ^bj L 5^' v! ^5 ^ ^^ i O-^"*^ * ** f^-l>^ (tfr*^ 
q* jus i^iXmJI j>^ *l3$ *rfl^ 4^5 ^^^ 'ft V^^' *^° 8I ^3 *** 

sJL* qma> *3 »^ob iJolX^ _£ *Jj**M ^5^3 xilll *xi5. V J^SJ soLiui 

^1 £*sM g^^ii <y*WI 3 tWU» 3 dJ^ I, J^» 3 <j-Jtf% c t^b 

oos of the adornments of wealth. Moreover, the only token by which 
men show a preference of some of the metals [over others] is their tech- 
nical use of the letter A, stamped npon any precious metal of which 
articles wanted are made ; and m regard to that they are controlled by 
the rarity of the occurrence of the metal, and the length of time that 
it lasts ; both which are distinctive characteristics of gold." But if, 
beside the rarity of its occurrence, and its durability, and the little 
appearance of moisture on it, whether moisture of water or humidity of 
the earth, or of its being cracked or calcined by any fire, and consumed, 
together with its ready yielding to the stamp, which prevents counterfeit- 
ers from passing off something else for it, and, lastly, the beauty of its 
aspect — if there is not [beside all these characteristics] some inexplicable 
peculiarity pertaining to gold, why is the little infant delighted with it, 
and why does he stretch himself out from his bed in order to seize upon 
it! and why is the young child lured thereby to cease from weeping, 
although he knows no value that it has, nor by it supplies any want ? 
and why do all people in the world make it the ground of being at 
peace one with another, not drawing their swords to fight, though at 
the sacrifice of the powers of body and soul, of family-connections, 
children, ground-possessions, and every thing, with even a superfluity 
of renunciation, for the sake of acquiring that ; and yet are ever 
longing for the thjrd stream,* to stuff their bellies with the dust? 

* There is here an evident allusion to the traditional saying : **>! ^SS ...( _^JL 

UfdfcJa u&A *Jl^ ^ XdaftJ^ v-*AX]l q* Q^y*J Ck*>!$ "and if the ton 
of Adam were to possess two flowing rivers of gold and silver, doubtless he would 
desire a third." 



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60 K. Khantkoff, 

[xf+>3 auuuol ^ b v_J&U ioyu vi^otf qI» «i qt^ ^'3 1^**^.^ '-^y -*^ 

y^ ^ IP^e ^ IAAjuj Jo j^XmmJS ^U>tf ^ XftAXtt ^ 

uX^oJ^ ^y^^ &e!yt cr* L $k 5=v ^!$ ^fcofl "^j *lS*N (3^ <>L*aJI 
O* -^ J^^ ^W^>y ^-^^ ^^ fr 4 ^ vJL>*fe X#aaftJt y.lfcu ^ or* 
juuw&j} JouaJI oIa^I^ KarUaX* jj-&3 q* sjkrf V^f} *<**■>" *$l^j 
v->Lj»t q* gUJ JUU^ ^a*3!j wJaJL w>i ja*?* U** u Ukc ^! taLoj ^UJL 

£ LgJU QjbCI itfiUjj Lp>bw vi^xis <A3 *lx&t LgJL* j$\Xj J^ Jafii vuXP 

Were it not for my fear of the physicians, I might also say that the 
soul's gladness at the sight of gold, the fine pearl, and the silk robe, 
falls little short of its delight in medicinal confections.* Nor does the 
soul take quietly the grinding up of gold, the pulverizing of the fine 
pearl, and the reducing of silk to ashes. It is only saddened thereat ; 
for, though by such means alone the heart be strengthened, yet men, so 
hearing, turn away from the [offered] exhilaration. 

Silver is next to gold as respects the peculiarities mentioned, and is, 
in like manner, made into tenders for things wanted and representatives 
of value for articles of necessity. 

Nor does the description apply only to fusible minerals. On the 
contrary, you may extend it to substances not fusible. Of these, the 
red hyacinthf [ruby] is equivalent to gold, on account of the rarity of 
its occurrence, the hardness of its crust, the abundance of its water, its 
lustre, the depth of its redness, its bearing of fire, its withstanding 
causes of injury, and its durability. Next to this come the yellow hya- 
cinth [topaz], that which is [blackish] like collyrium, the emerald, and 
the chrysolite, which differ [from the ruby] and are equivalents of 
silver. To all the above named the fine pearl is manifestly inferior, as 
appears from the softness of its body, its being generally composed of 

* Oar author evidently alludes, in jest, to the famous *»<** ^y^MA , "exhila- 
rating confection," of the oriental physicians, 

f For want of a better word, we use " hyacinth" in a generic sense in this chap- 
ter, to represent the Arabic OjSb , as applied to all precious stones alike, a word 
which has no proper equivalent in any European language. See D« Sacy's Chresi. 
Ambe,2<fceUiii.4d4. 



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Book of the Balance of Wisdom. 61 

y-ij*& "* 1* o U J *■* o^!; £ j*N* k 31 ** 1 o* ! l^" 1$^ u* 1 -^ l5^' 

Uit^ cX^fc!! ^j^Sj u**a^ ****>■ ^3 v^iLit J***! h^j {J* *^' ^^ 

j L y3" £*^>* i£& o^ccult oUxcJl^ AjLSt q* J$& *JJij &&> OuSoJ 

v>*»^£ cUAoSf *JLc y^ Lfc^' q» Hj^iL o^y^li Lf^ju yjoi^ ^LH 
\J}j*l\ y?\JL it v^L^i ffr»^ J N^>2> **^ J*** ^ !****« ^ 

^l^t ^Uj JLil s^^' a* U^! ^ ^wbuu L** ^^liu D \ ^^X*^ 
CT* ^H^ 1 V*"** (!>*** o* ^J** ^ cy**^* *****' Ifcv* £**> ^ \J&) 

pellicles, doubling one upon another, like the coats of an onion, its being 
reduced by fire to ashes or rotten bone, and its change of color from the 
action upon it of medicine or perfume, or other like causes of deteriora- 
tion. Yet one finds no fault with its price, nor at all undervalues it. 

The number of the precious stones is not thus exhausted. But suffice 
it to say, on the other hand, that certain gems are mentioned, of which 
the mines are no longer found, and the specimens once in the hands 
of men have disappeared, so that people are now ignorant of what sort 
they were. There appear, also, from time to time, gems not before 
known, such as that red gem of Badakhshan, which, were it not for its 
softness, and that the water of its surface lasts but a little time, would 
be superior in beauty to the [red] hyacinth, and is no antiquated gem. 
The mountain containing it was fissured by an earthquake, and the 
windings of the rent brought to view, here and there, egg-like lumps of 
matter deposited in layers, resembling balls of fire, of which some were 
broken, so that a red light gleamed forth beneath where they lay. 
Lapidaries stumbled upon the gem, and gathered specimens, and, hav- # 
ing nothing to guide them respecting [the purification of] its water, and 
the polishing of its face and making it brilliant, were, after a while, led 
by experiments to make use of the stone called, on account of likeness 
of color, the golden marcasite, and with that succeeded ; and the mine 
has yielded abundantly. 

It is not impossible that both fusible and infusible substances, now 
unknown, may be brought to light, at any time, from the undercliffs of 
mountains, and from the beds of rivers, the depths of seas, and the 
bowels of the earth. In respect to such, however, we will not barter 
away ready money for a credit, nor turn from the known for the sake of 



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62 K Khanikoff, 

{ ^\ ^Lxb! KlsJ jIj^ Jtf] Ju otjlftfl JjLsj U^ x>j^t ^sJuj /^ 

^xmu oj* Jc Lf l^ y>^S. V L^ J jrflwJ' SU^U ^ ^L^Cf ^ ffy> 

LoU^ ^U> ^3 1^*1 ****** o^ **' i^ 5 * »>l**^M y.L* jl>° k^jsJ^ 

the unknown. For I am interested in the subject of which we are in- 
vestigating the rudiments, and endeavoring to get at the main supports, 
And to tread upon the foundations, on account of the counterfeitmgs of 
those who put in circulation vitiated and spoiled money, and practice 
adulteration in the goldsmith's art But precious stones are counter- 
feited as well as metals, nay, oftener and more successfully, because the 
eye is little accustomed to them, and wants the guidance of habit in the 
choice of them ; while, on the other hand, no one in any city fails to 
see dirhams, and no trading is possible without the handling of dinars. 
The possessors of precious stones and jewelry are few in number among 
men, while those who make use of metals are ever to be found. We 
must, therefore, treat of precious 6tones as we have treated of metals, if 
the Supreme God so wills. 

Sboxdon Fibst. 

Statement of the Results which toe have obtained, by the [Conical] Instru- 
ment, in respect to Precious Stones. 

We will first enumerate the precious stones which have been com- 

Eared, and afterwards exhibit their proportions [of weight], as proved 
y comparison. 
1. Hyacinths. When the common people hear from natural philoso- 
phers that gold is the most equal of bodies, and the one which has 
attained to perfection of maturity, at the goal of completeness, in re- 
spect to equilibrium, they firmly believe that it is something which has 
gradually come to that perfection by passing through the forms of all 



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Bock of the Balance of Wisdom. 63 

ULJt «53J Juu JL*2 toy ^ Ui ^ l^Uc* dj*?. (£ \jji Qjik a 5 

vju^j? j*ob=*ji s^toii £*&u <y ^^ -5' cy^' 03^ u ^ ***** 

[other metallic] bodies, so that its gold-nature was originally lead, after- 
wards became tin, then brass, then silver, and finally reached the per- 
fection of gold ; not knowing that the natural philosophers mean, in 
saying so, only something like what they mean when they speak of man, 
and attribute to him a completeness and equilibrium in nature and con- 
stitution — not that man was once a bull, and was changed into an ass, 
and afterwards into a horse, and after that into an ape, and finally be- 
came man. The common people have the same false notion, also, in 
regard to the species of hyacinth, and pretend that it is first white, 
afterwards becomes black, then dusky, then yellow, and at last be- 
comes red, whereupon it has reached perfection ; although they have 
not seen these species together in any one mine. Moreover, they 
imagine the red hyacinth to be perfect in weight and specific gravity, 
as they have found gold to be ; whereas we have ascertained that the 
celestial species [sapphire] * 3 and the white [the diamond] exceed the red 
in gravity. Of the yellow I never happenea to have a piece sufficiently 
large to be submitted to the same reliable comparisons already made 
with other species. 

2. Ruby of BadakheMn. I have, in like manner, never obtained such 
a piece of the yellow species of this gem that I could distinguish it 
from the choice red, called piyazakt, that is, the bulb-like. 

3. Emerald and Chrysolite. These names [^/^' and uX^-^J!] 
are interchangeable, whether applied to one and tne same thing, or 
to two things of which one has no real existence ; and the name of 
emerald is the more common. I have, however, seen a person who 
gave the name of emerald to all varieties of the mineral excepting the 
beet-like, or basil-like, 14 which has an equally diffused green nue, and 
is perfect, transparent, and pure in color ; and who denominated the 
latter chrysolite. 



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64 N. Kkanikoff, 



L5 15 ^ i>-!; J j^ cr IO*A^ ^ !**->*<• o^ ^ o'j g^'* j^> 
,aJ&XM j\&*W **0 ^35^5 sUjAAfil ^CJAli ^jJlJI »Uit q* «13 
*Laj &>>>} &3 ^' f^AxJS ^c Uju».j >J Lg£L£t} s»"»4»2 gjJj**^ 
J(j ak>ta cr *cji cr u^J U L; .to iu£L&4 ^ gjdj**H o^ fckx *^ 

**« ^ijitJ^ gj^' ls*** cr^ 1 cr l^ j^ ^^^ cr y/*J' ^ 

iujji y> X*Lit ^5^.^ ^hV^- u*y* ^ d**+~* ^ <y^ UuiuJl 
^^1 *uUju ^ D t^> (J^^ W5 xJLJl *xp c^^BI ltaJ^W 

JLSi\^ L(ju ,U*J! e^ vtt^ jsJt tulsLrfuilj otyjjlfe ^iib fc>,lii 

4. Cornelian, Onyx, Lapis Lazuli, Crystal, and (Now (this last — al- 
though it is not the product of a mine, but, on the contrary, kindred to 
stones, or sand, or alkali — because it resembles crystal, for which reason 
we have submitted it to comparison), and precious stones similar to 
these, such as Malachite, Turquoise, Amethyst, and the like. The mala- 
chite [itself], on account of the rarity of its occurrence, from there being 
no mine of it now known, has been unobtainable ; so, too, the turquoise, 
which, besides, always has within it a mingling of foreign matter. This 
whole class of stones is not highly prized ; excepting the onyx, for a 
certain value is attached to specimens of this mineral marked with 
ox-hoof circles, 1 6 and likewise to those in which there happens to be 
presented the form of an animal, or some strange shape. Men have 
been Jong tired of the cornelian, so that it has ceased to be used as a 
stone for seal-rings, even for the hands of common people, to say 
nothing of the great The lapis lazuli is employed on account of the 
tinting and variegation of its several species. 

5. The Fine Pearl. The pearl is not a stone at all, but only the bone 
. of an animal, and not homogeneous in its parts. Yet I associate it with 

the hyacinth for its beauty, as I join therewith the emerald both for its 
beauty and its rarity. It therefore comes in here with as good reason 
as they do. Besides, there is no such difference of opinion respecting 
the minerals which have been mentioned, as exists in regard to the 
water-equivalents obtained in the case of pearls ; nor have the acces- 
sions or losses, as between small and large ones, been recorded — a point 
on which there is great diversity. What I shall state, as to the pearl, 
applies to those which are large, full, and rounded. 



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Booh of the Balance of Wisdom. 



65 



n>**^ ^>U^ *& *** j^»oL« kS^z ^^' vXiiXfej Jjj-jASa* j^fi 



A*, 



1.1 

irf 

1v. 

*ir 
irf 

in 

m 
11. 

11 f 



tfLJt *XP -pjLmJ^ 



^Lw03JULotjv> 



SlftJu* 



**** * 

c 



o^r? **4ji 8 I^**j ^ ^ 



,ua 



o 



L*~ 



LSI 



a' 



^■i 



L5~ 



* 



L*~ 



iX>l< 



L$~ 



K m*£ 



LA 



o' 



t\D»t, 



C^" 



*V* 















OwJt 



6. Coral. This is a plant, though petrified, like the Jews' stone 1 6 and 
the sea-crab. 1 7 There is a white species of it, coarser than the red, per- 
forated throughout, and divided; which I have not compared, because 
men use it but little, and also because I have heard speak as if the red 
were white when torn off, and became red by exposure to the air. 

We have put together the following table. 

▼OL. TI. _ . 9 



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66 



JT. Khanikoff, 



x* 3 L»4 L^ tXp»t» JT xi> D J JLfi ^iUt K^UsJt y>!^<y *&■ 

JjjJl j*» <y UT X*»L*U iUi^lft jUeffl Jo!,* *i o»^ ^JJU 

Ta5fe o/ Weights of the Water-equivalents of Precious Stones, 
supposing all the Weights in Air to be a Hundred Mithk&ls. 



Namei of Predona 
Stonea. 

Celestial Hyacinth, 

Red Hyacinth. 

[RubyjofBadakh- 
shin. 

Emerald. 

Lapis Lazuli. 

Fine PearL 

Cornelian. 

Coral. 

Onyx and Crystal. 

Pharaoh's Glass. 18 



Weights 


of Water-equir- 


IGthkal&DflJuka 
25 1 


Taasuje. 
2 


26 





27 5 


2 


36 


2 





37 


1 





38 


3 





39 i 





39 ! 


3 


40 








40 


1 






Wftter-equiralents re- 
duced to Taaaaje. 

Six hundred and six. 
j Six hundred and 
( twenty-four. 
( Six hundred and sev 
I enty. 

( Eight hundred and 
{ seventy-two. 
(Eight hundred and 
\ ninety-two. 

INine hundred and 
twenty-four. 
iNine hundred and 
thirty-six. 

(Nine hundred and 
thirty-nine. 
{Nine hundred and 
sixty. 
iNine hundred and 
sixty-four. 



Taaaajam 
Numeral* 



606 
624 

670 

872 

892 

924 

936 

939 

960 

964 



Siotioh Second. 

Relations between Weights of Precious Stones alike in Volume. 

By way of correspondence with the computation already given of the 
weights of equal masses of metals, a similar estimate is [here] furnished 
relative to precious stones of like volume, supposing that each mass is 
equal in volume to a hundred mithkals of the collyrium-like hyacinth ; 
in order that one who would ascertain any proportion {of weight] re- 
quired may be enabled to do so, through the properties of four mutu- 
ally related numbers. 



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rf.. 
rivi 
iir. 

fovf 
IdfO 



f aw 
Jot, slsj a UJ! 

1 0.1 MUwJjSUvmA^jUJ! 



Aoi of the Balance of Wisdom. 07 






£•».!» ,.,uji 






wis 



a 



.us? 



O 






o 



J^!, 



J^>f, 















a U1 



Ztofc o/ TFeiyAfr ofPreckm* Stone* alike in Volume. 



vJUSUtt 
JwJt 



Karnes of Pre- 
cious Stones. 



Celestial _ 
Hyacinth. " 
Red 

Hyacinth. J 
Ruby [of Ba- 
dakhahan]. 

Emerald. 
Lapis Lazuli. 



Weights, when the Vol 
ume w equal to a Hundred 
Mithkals of the CoUyrium 
like Hyacinth. 

Tajsujs. 




Mithk&ls.|Daoiks. 




100 
07 
90 
69 
67 




2 
3 
5 



8 
8 

2 



Reduction to Tassujs. 



Two thousand four hun- 
dred. 
Two thousand three hun- 
dred and thirty-one. 
Two thousand one hun- 
dred and seventy-one. 
(One thousand sii hundred 
and sixty-eight 
' One thousand six hundred 
and thirty. 



Tassujs in 
Numerals. 



2400 
2381 
2171 
1668 
1630 



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68 



N. Khamkoff, 



wen v^otf w d^n, *ut <y wb&u /j^L v* j*^ vK ^ w 



Weights, when the Vol- 
Karnes of Pre- ume i» equal to a Hundred 
does Stones. MithkAlsofthefoUyrium- 
like H jacinth. 

. Taeaujs. 



Fine PearL 
Cornelian. 

Coral. 

Onyx and 
Crystal. " 
Pharaoh's; 
Glass. 



[TABU OONTUIUKD.] 

Reduction to Tassajs. 



Mithkala. 
65 

64 

64 

63 

62 



Dftnike. 
3 

4 

3 



5 



2 
2 
1 
3 
1 



iOne thousand five hundred 
and seventy-four. 
IOne thousand five hundred 
and fifty-four. 
[ One thousand five hundred 
! and forty-nine. 
; One thousand five hundred 
! and fifteen. 

; One thousand five hundred 
! and nine. 



TaesdJB in 
Numerals. 



1574 
1554 
1549 
1515 
1509 



Section TmfcD. 



Relations of Air-weight* to Water-weighU. 

We resort again to water and the just balance, and propose thereby 
to ascertain the measure of the difference between the weight of any 
one of the several precious stones in water and its weight in air. When 
the bowl containing the precious stone is once in the water, that is 
enough — you thus get its weight in water, after having weighed it iu 
air. This is a great help to a knowledge of what are genuine precious 
stones, and to their being distinguished from those [artificially] colored. 
'Abu-r-Raihan does not speak of this matter, but at the same time his 
statement given in the first section of this chapter facilitates the settle- 
ment of it ; that is to say, we may take the weight of its water-equivalent 
Saere] stated, for each precious stone, and subtract it constantly from 
e hundred mithkals constituting its air-weight, and the remainder will 
be its water- weight. 

Now we have set down these water-weights in the following table. 



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Book of the Balance of Wisdom. 69 

^jJ.] *>\ } xJt,* JlfiU 8^ x^UI Jjj» Js^s- 






Mf 
Jvvl 

wr. 

IoPa 

Id. A 

ffvl 

Iflf 
Iflt 
Iff, 

ifn 



_ * I 

&u,t 3 bL*ju,I 3 vJJl 
cx»-l 3 HUa*,! 3 v_aR 



*JUi l^t 





vJuit^O 


o^ 


**$.! 


^K 




a US! 


L^^ 


^yili 


AJU^ 


**N 


"lUftj^- 


C5*" " 




U&V 


*&ds 


^"i 


^yiK 


Ju>t 3 




L^*^ 


* 




L^ * 





JUU.t 

a Ufl 






TaWe qf Water-weights to a Hundred MithkaU in Air, 
added by 'al-Kh&zinl. 



Names of Precious 
Stones. 



CeleBtial Hyacinth. 

Red Hyacinth. 

[Rnby]ofBadafch- 
sh&n. 

Emerald* 
Lapis Lazuli 



'Water-weights. 


SDtbkflU. 


Diaiks. 


Tuifip. 


74 


4 


2 


74 








72 





2 


63 


4 





62 


5 






Reduction to Tassujs. 

f One thousand seven 

j hundred and ninety- 

' four. 

r One thousand seven 

j hundred and seven- 

' tv-six. 

( One thousand seven 

i hundred and thirty. 

(One thousand five 

j hundred and twenty- 

' eight. 

J One thousand five 

( hundred and eight. 



TastujaiB 
Numerals. 

1794 

1776 

1730 
1528 

1508 



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70 



N. Khamkoff, 



ju^I li v^^K ^ oJl> JI ^ .ALiOj j^« o^Ltf »W- ^ i5sl» 

(YASLB OOMUMUB0.) 

Redaction to *ftwtyi. 



Name* of Precious 
Stones. 


Vat 
MUhkftb. 


er-weig 
Diniki. 


hts. 
Twauji. 


Pine Pearl. 


61 


3 





Cornelian. 


61 








Coral 


60 


5 


1 


Onyx and Crystal 


60 








Pharaoh's Glass. 


59 


5 






{One thonsand four 
i hundred and seven- 
' tv-six. 

rOne thousand four 
« hundred and sixty- 
(four. 

rOne thousand four 
< hundred and sixty- 
lone. 

5 One thousand four 
c hundred and forty. 
{One thousand four 
j hundred and thirty- 
'six. 



Turijsm 

Numeral*. 

1476 



1464 

1461 

1440 
I486 



Saonoir Focbth. 
Instruction and Direction relative to Difference of WaUr-tquivalents. 

There is not the same assurance to be obtained in regard to these 
precious stones; as in regard to fusible bodies. For the latter bear 
to be beaten, until their parts lie even, which expels the air that may 
have got into them in crucibles, and separates them from earthy mat- 
ter. Moreover, we know not what is in the interior of stones, unless 
they are transparent, and can be seen through (for, in that case, what- 
ever is within them appears), so that doubt has arisen in my mind as to 
the lightness of the red hyacinth, and the difference in weight between 
it and the dusky species. For, both the dusky and the yellow being 
very hard, no earthy matter, or air, or any thing else, mingles with 
them ; which is rarely the case in respect to the red, inasmuch as most 



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Book of the Balance of Wisdom. 71 

slJ^ ^11% jJ^aJb U4J5 s3j ^if y^ J*-^ *Lfi^t JU* g*X% ^ 

^ «y Uu, »^ * e^^ c^ g^ t *I^U Uu^bi* t^ «>Ui Jf 

pUg^ U^lf xJUli «I L^b ouc*Ll! LfLMX^ a t a ftv tt ^U9 3 l 
AJJ^j xU^ui! w^J «*Jtf <$& *** L*i Jj *«**£ ^U> ^ *ltt 

V^y CT^ *^ u^i<-V vjaAJu. qI ^ ji? jui^ KJb> £tot>* *ilU0 
ij&£»t q. L|aU *jja& ^Ult 3 Jj.UU 3 g£ Ult Kf> cr k^ j6*2f jJt 

U £&*^> r^ 1 " >i W IfcO *fjfSt JL^. 1 g I » &** ftu^t Jj^aftJ! £>LfcJ? 

specimens of this species nave bubbles within, fall of air, or, being 
mixed with earthy matter, are not without air on that account. Nor is 
the red hyacinth so splendid in color when first gathered, until fire, 
kindled upon it, has purified it ; and, as it becomes hot, whenever there 
is air in the gem, it swells and is puffed up, and bursts, in order to the 
escape [of the air]. People, therefore, bore into this gem, by means 
of the diamond, opposite to every bubble or particle of dirt, to make 
way for the air, that it may escape without injuring the gem, and to 
prevent a violent and rupturing resistance to expansion. When such 
borings are not made, or are too small to allow of water entering into 
them, on our immersing the gem in the [conical] instrument, the quan- 
tity of water displaced is not precisely in accordance with the volume 
of the gem, but, on the contrary, is as that and the penetrated air* 
bubbles together determine. In like manner, when the emerald is 
broken, seams appear within, or, in their place, some foreign matter is 
found. Possibly, empty cavities always exist in this mineral. But its 
rarity prevents any diminution of its price on that account 

Whoever looks into our. statements, and fixes his attention upon our 
employment of water, must be in no doubt as to well-known particulars 
concerning waters, which vary in their condition according to the reser- 
voirs or streams from which they come, and their uses, and are changed 
in their qualities by the four seasons, so that one finds in them a like- 
ness to the state of the air in those several seasons. We have made all 
our comparisons in one single corner of the earth, namely, in Jurjantyah 
[a city] of KhnwArazm, situated where the. river of BaUth becomes low, 



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72 N. Khantkoff, 

s^ iV*y <y «*^> ^^i*. * 05^ 4i*> *>* ^ ****** ^ 

J^ J^J! r b U li^aj (j-JiS b^ jt lu^ D tf s\y»} *U*Jt q* OM/^ 
4« & ju&u ULfi^i Jj *U! jfjk Llii JujJ^ ^1 ^ ^t Aio vX^-1^ vJuao 



eJUJJ V M 

^lj^3 otjUH ^.g^ *L£< 0^ ^ 

crfjUll (ja« or *LU tfU. Ju^. qJ aJJ* v^3Jt >Xtt1 ^ <3*^ 

at its outlet upon the little sea of Khuwarazm,* the water of which 
river is well known, of no doubtful quality ; and [all our operations 
have been performed] early in the autumnal season of the year. The 
water may be such as men drink or such as beasts drink, not being 
fresh :f either will answer our purpose, so long as we continue to make 
use of one and the same sort. Or we may use any liquid whatever, 
though differing from water in its constitution, under the same limita- 
tion. If, on the other hand, we operate sometimes with water which 
is fresh and sometimes with that which is brackish, we may not neglect 
to balance between the conditions of the two. 
This is what we wished to specify. 

* This k positive testimony that, already at the commencement of the twelfth 
century, the Oius no longer emptied into the Caspian, hot into the little sea of 
Khuwirasm, that is, into the Sea of Aral. In order to contribute to a complete 
collection of those passages of oriental authors which relate to this interesting fact 
m the geographical history of our globe, I will cite a passage from Kaswinf s 'Ajd'ib 
W'MakhlSkdt, referring to the same met in the following century. ' In speaking of 

the Jaihun, this author says : ^ fj)\^ v)^^ iS*^" 8 /A*^ cf** ci* J 4 * £ 

1^1 &u« (•);!•£• ttfjfe I*** rj;^ 8 j** vs*"* 8 ^ 5? * ^ 4***** " u £*• 

Jaihun] then passes by many cities, until it reaches Khuwirasm, and no regiom 
except Khuwarasm profits by it, because all others rise high out of its way ; after- 
wards it descends from Khuwarazm and empties into a small sea, called the sea 
of Khuwarazm, distant three days' journey from Khuw&razm." See el-Caxwinf a 
Kosmographie, ed. Wiistenfeld, l r Th M 177.19 
\ The Tibydn, a commentary on the Kdmfa, thus demies the two terms yu |_*» 

and v^-^ : *A* ^ J^ u*L*M *4y**A ^5 Xj^uXc a«o ^mJ ^jSS\ \ r *jJxZ\ 
*jyXj %X*} B^yasi! A*£ "it (j^Uit **f&$2 *^vXa jus <j*J ^o^XJJ ^f~^^3 
ajL&JI u water called tharib is that which is not fresh, and is drunk by men just 
as it is ; and that called thartib is water not fresh, which men do not drink except 
from necessity, but which is drunk by beasts." 



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Booh of the Balance of Wisdom. 



73 









Chapter Third. 
Observation of Other Things than Metals and Precious Stones. 

We are [now] led to [consider] the proportionate weights of wax, 
pitch, resin, pore clay, enamel, amber, and woods of well known trees 
— being the materials of models and patterns formed by goldsmiths, or 
others practising their art — for the sake of any one who may wish to 
cast an equivalent weight of some metal, after the goldsmith has pre- 
pared, by his art, a pattern [of] known [material and weight] ; including 
also the proportionate weights of other substances necessarily or option- 
ally made use of. We have set down all these substances, with their 
water-equivalents, and their weights [in equivalent volumes], in two 
tables. Let, then, the water-equivalent be measured by the [proper] 
table, and by that let the proportion of metal sought for be determined. 

Here may be diversity of opinion — to every one his own ! 

Tliis chapter has two sections. 

Section First. 
Knowledge of Weights of the [ Water-equivalents of] Materials of Models, 

when (he Weight obtained out of the Water is a Jfundred Mithkals. 





vjuitjs} 


J*3&. 


*Utfl 

ki 


ir*A 


t 


V—i 


o 


^L^U^vJt^viait 


l.lf 


V_l 


s 


M 


>-n "& 


mf 


t 


t 


o» 


&*M*J) 


t**t**yA 


V-J 


£ 


r* 


{jM2%i\JL«*Ji 


fArr 


t 


I 


tt 


k^» 


it. 


V— 1 


v-J 


** 


LuJJ 


rn. 


V— 1 


'» 


_yO 


^sJt 



Names. 


Mithkah. 


D&niks. 


Taasujs. 


Reduction to Taseute. 


Floating and 
Sinking, so 


Clay of Siminjan. 


50 


2 





1208 


s. 


Pure Salt 


45 


3 


2 


1094 


« 


Saline Earth. 


90 


1 





2164 


a 


Sandarach. 


140 


4 


2 


3878 


fl. 


Amber. 


118 








2832 


u 


EnameL 


25 


I 


2 


610 


8. 


Pitch. 


96 


2 


2310 


« 



VOL. VI. 



10 



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74 



K Khamkoff, 






roff 
Iflf 

nrf 

11a 
Yotf 
oloo 



o l 






tfrv 

1.1a 

n.f 

tlAV 

iff. 
trrr 
irr. 



i" *|jt ^ oljuUl yj^ A9JM ^ 

' •" *u«ut 



£ 


1 


ii 


t 


i 


U 


fc 


z, 


e* 


I 


V 


f 


t 


\_» 


j» 


Z ' 


k 


t?> 



tr.. 

Ho. 

trfi 

Ml 

trrt 
tirr 






*UaJI *U 



[TABU OOXTINUBD.] 



Names. 

Wax. 
Ivoiy. 2 * 
Black Ebony. 
Pearl-shell. 
Bakkam-wood. 
Willow-wood. 



Mithk&li. 


DAnika. 


Twefiji. 


105 


1 





61 








88 


3 





40 


2 





106 


2 





248 





3 



Redaction to Taesajg. 

2524 
1464 
2124 
968 
2552 
5955 



Moating and 

Sinking. 

fl. 



Section Ssoovd. 



Knowledge of Weights of Liquids in a Vessel which holds twelve hundred 
[of any measure] of Sweet Water. 



Name*. 
Sweet Water. 
Hot Water. 
Ice. 

Sea-water. 

Water of Indian Melon. 
Salt Water. 
Water of Cucumber. 
Water of Common Melon, 
Wine-vinegar. 



Weight*. 
1200 
1150 
1158 
1249 
1219 
1361 
1221 
1236 
1232 



Names. 
Wine. 

Oil of Sesame. 
Olive-oil 
Cow's Milk. 
Hen's Egg. 
Honey. 

Blood of a Man in good health, 
Warm Human Urine. 
Cold 



Weight*. 
1227 
1098 
1104 
1332 
1242 
1687 
1240 
1222 
1230 



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Booh of the Balance of Wisdom. 75 

\JOj\l\ s&*j\\X8a} ofjia% *UI y+ j*£a cK<3 .LXfit^ s\\\ (j*LJU ^ 

a>UI jJtAiUJ JliiSUt w^*J J^uoagj *ltt (j^Liu ^ 

jus *IU Cuwai UAX>t qJ^ ^Lftxit ( JLa a** UV^j «■*** *£^.5 (V 

^yt gl£*J» ^'5 (jmJU^ Uli&o cfcft**^ c^b H ^^ ^ *ip °^ 
^i! Lp\Xo ilb cXS SuflJL> &&39 Ja*r> .J! iVXc^ y^CJ! *L& fc>L»^ 
lftSl^»t ^ v!^l ?}j* o* ^ y^ ***;' J^ 31 ^ KS13 ,X ^L» t 

i\& «JbASait Ut^ 1 taj^> qj.w,^ iuuwc^ n^^° L5E>^' O* ***£ l* r& 

Chapter Fourth. 

Device for Measuring Water, Comparison between a Cubic Cubit of Water 
and the tame of the Metals, and Quantity of Gold sufficient to fill the 
Earth. In Three Sections. 

S«cnoir Fibst. 

Device for Measuring Water, in order to the Determination of Relations 
between Heavy Bodies, on Premises of Superficial Mensuration. 

'Abu-r-Raihan ordered a cube of brass to be made, with as much 
exactness as possible, and that it should be bored on its face, at two 
opposite, angles, with two holes, one for pouring water into it, and the 
other for the escape of air from it ; and he weighed it in the flying 
balance, first empty and hollow, then filled with fresh river-water of the 
city of Ghaznah ; [and] 392 mithkals and £ and £ of a mithkal [proved 
to be the weight of that water which it would contain]. Wanting, now, 
to get the superficial measure of one [inner] side of the cube, he had 
recourse to a thread of pure silver, so finely drawn that to every three 
mithJjAls [of its weight] there was a length of fourteen of the cloth-cubits 
used in clothing-bazaars. He trimmed off from the length of a side [of 
the cube] the thickness of two of its opposite surfaces, and wound the 
thread around the remainder ; and what this would hold of the thread 
wound around it, was 259 diameters. Now, the [length of a] side of 
the cube [thus shortened] would go into a cubit four times, with a re- 
mainder which would go five times into that length, leaving a [second] 
remainder which was one-ninth of that length. The [lenjrth of a] side 



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" ^ 



76 K Khanikoff, 

(jyuj!^ jUmJ? **«iut cXS fi*5fi3t im? ft^^** ffH^ £**^ vfi^Jtf ajJIj juJu 

H^iXit J^-Ji Qt f^A" ***^ jf^' £*^ ^$* Kmm3- ool^ j^t ^aJ( 
& "vSyB* <AJ3> ^Xs>yf cfc**j!» **»+*> cr |p5 o-^Mj o^ ^ 
V^t&oj **f a11P LLJ Ck**;!$ <j**^ J^-^*- j-y^ *j/° Cfc**;^ i U* ^> 

• 4 »Lo yfo (iUJj (j*mJ*5 vi>i^ r4 j*^ £«mJ > gsjL*JaJ\ Qt (^^b FaI.o 

of the cube [thus shortened] was therefore understood to be divided 
into forty-fifth parts, of which the first remainder, the excess of a cubit 
above four times that length, made nine forty-fifths, and the second 
remainder of that length, (of which the first remainder was one-fifth,) 
made five forty-fifths, which is the same as one-ninth of that length. 
Consequently a cubit would take in 1082^ of the mentioned diameters 
of the thread ; which being multiplied by 45, 48,692 is produced as the 
[number of] diameters of tne thread to forty-five cubits. 

The cube of the [number of] diameters in the [shortened length of a] 
side, namely, 259, is 17,373,979 ; and the weight of water of the same 
volume is 9415 tassujs. But we have said that the number of diame- 
ters of the thread to a cubit was [found to be] 1082^, of which the 
cube is 273,650,180,698,467 [-^60 3 — 216,000J. So then, if we multi- 
ply [this sum] by the [number of] tassujs of (water contained within] 
the Drazen cube, and divide the product by the third power of [the 
number of diameters of the thread held within the length of an inner 
side of] this cube, the quotient is the [weight in] tassujs of a [cubic] cubit 
of water, namely, 686,535 and about £ and £ more. If we divide this 
weight by 24, the result is in mithkals, of which there are 28,605, with 
a remainder of 15 tassujs and £ and |. That is the weight of a cubic 
cubit of water. The fractions in this sum are consolidated [by multiply- 
ing it] into 360 ; which gives [the weight of] three hundred and sixty 
cubits cube [of water], amounting, in mithkals, to 10,298,033. 

This is what we wished to explain. 2 a 



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r^ 



Book of the Balance of Wisdom. 77 

jU Jf a* y^CJI ^JJI D !^ ,xx* ^ ^ 

Jl&ftl £vj U Jytolij ^ y>! ^o\ Ji\ v>yu J*dfl !Jtf> Aya&» cUc^ 

*W cy3 ^ 5*B *UI a j3 iU^S ^^ L^Im »LJt ^ a ^Jj £ 
iu~i a b Jjrtl *Ut ^3 d> ^t jSrtl *m ^ o) > x^jSsM 

8U vX q^ LT-^S V&*^> L>jj*bb j&& t Lm+5* LfJtyU FaI.o j»SUN 
j»SI\ ^JJi *L*f v^otf lfy> yjsL^ *\^yuto ^ lit ^U&o efciLSj efc^ 

Sxorioir Second. 

Knowledge of Numbers for the Weighte of the measured Cubit of all 

Metals. 

The principle last considered having heen made out, we turn to an- 
other, which is a difference Tin weight] between heavy bodies of like 
masses, but differing in kind, by virtue of relations subsisting between 
metals in respect to volumes. We have already stated, in the first chap- 
ter of this lecture, that whatever may be the relation between heavy 
bodies alike [in volume], as to [absolute] weight, is known from their 
water-equivalents ; and that the relation of the weight of the less water- 
equivalent to the weight of the greater water-equivalent is as the relation 
of the weight of that body of which the greater quantity of water is the 
equivalent to the weight of that body of which the less quantity of water 
is the equivalent. Consequently there must be an inverse relation be- 
tween tne [absolute] weights of heavy bodies and the dimensions in 
length, breadth, and height, of those water-equivalents put down. 

Now for a second principle. Since the weight of a volume of water 
equivalent to the cube of the measured cubit is 28,605 mithkals, together 
with 15 tassujs and £ and -J, and since 182 mithkals make a mann Tone 
mann being computed at 260 dirhams}, [a cube of] the measured cubit of 
water weighs 157 manna, 6 'istars ana i and J and -J. It is also known 
that the weight of the [cube of the] measured cubit of any metal what- 



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^i 



78 



N. Khanihoff % 



^j^uJj jk!i i*Uj ^Ai tfoUi) &j>J> otjiilt i*Uj «L* er *xH* *K J^ 
jls JT cr j~£' £^ ob* 1 ^ 

vi>Jl3 J. 111a ^j*mJ*5 v£aJL> 

oUtt 

ever is to the weight of its equivalent of water as 2400 tassujs of that 
metal to the [weight in] tassujs of its water-equivalent, put down oppo- 
site to it in the table [above given]. The first of these proportionals 
being unknown, if the second is multiplied into the third — I mean, the 
weight of a volume of water equivalent to [the cube of] one cubit, which 
is, in tassujs, 686,535, being multiplied by 2400 — and if this [product-] 
number is divided by the [weight of the] water-equivalent of each of 
those metals, the quotient is the weight in tassujs of a [cubic] cubit of 
that metal. 

It will do no harm to put down, opposite to each metal, the weight 
of [a cube of] the measured cubit thereof, in mithkals, tassujs, and 
fractions of tassujs, and the number of manns and 'istars which that 
amounts to, in a table, as follows : 

Numbers for the Weights of the measured Cubit of all Metals. 29 



/* i ' u 


*U«! 


* 


nir 


} 


nn 


0*4 


Ivv1 


1 


IIIa 


\ 


rm 


z 


Iffv 
Mo. 





MM* 


ol^UiUUw! 


L 


offAll 


v^xJJ 





t^AVAvP 


v-***jtt 


yJ 


rr^Arv 


o^*il 


kS 


Mflo. 


MttJt 


3 


PfvAfl 


(jwLsajJt 


PfoHl 


JU^Jt 


\ 


mfir 


OufX&l) 


^S> 


f«1M 


o>U>jJl 



Names of Metals. 


Mithkals. 


Tassujs. 


Fractions. 


Ttf^yif)^, 


Tatars. 


Gold. 


544,869 


11 


i 


2993 


31 


Mercury. 


387,873 


4 


i 


2131 


6 


Lead. 


323,837 


12 


i 


1779 


14 


Silver. 


294,650 


10 


i+i 


1618 


38 


Copper. 


247,846 


18 


i 


1361 


31 


Brass. 


245,191 


6 


i 


1347 


8 


Iron. 


221,463 


1 


i+i 


1216 


33 


Tin. 


209,309 


14 


f 


1150 


2 



Fractions. 



+i 



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Booh of ike Balance of Wisdom. 79 

This exhausts all the more interesting; matter which admits of 
being extracted from the work now under analysis. In the sec- 
tion following the last translated, our author sets himself to calcu- 
late the quantity of gold which would compose a sphere equal to 
the globe of the earth. He prescribes to himself this task almost 
as a matter of religious obligation, in order to find the ransom 
which, according to the Kuriin, the infidels would offer to God 
in vain for the pardon of their sins ; for he begins with citing 
the eighty-fifth verse of the third chapter of the Kuran, which 
reads : " truly there will not be accepted as ransom from those 
who were infidels and died infidels as much gold as would fill 
the earth ; for them there are severe pains ; they shall have no 
defender." We will not follow the author in his laborious calcu- 
lations, but will content ourselves with merely noting some of 
his results. He says that the cubit of the bazaar at Baghdad is 
twenty-four fingers long, each finger being of the thickness of 
six grains of barley placed side by side. The mile contains four 
thousand cubits, ana three miles make a farsang. The circum- 
ference of the earth is 20,400 miles, and its diameter is 6493£f? 
miles. Finally, the number of mithkfils of gold capable of fill- 
ing the volume of the globe is, according to him : 

36,124,613,111,228,181,021,713,101,810. 

For the purpose of comparing these numbers with ours, I will 
observe that the radius of a sphere equal in volume to the 
spheroid of the earth is 6,370,284 metres ; this would give us 
one mile = 1962.048 m., and one cubit = 490.512 millimetres : 
that is to say, if these measures admitted of a rigorous compari- 
son ; but Laplace has very justly observed* that the errors of 
which the gebdetical operations of the Arabs were susceptible 
do not allow us to determine the length of the measure which 
they made use o£ for this advantage can only be the result of 
the precision of modern operations. I have endeavored to meas- 
ure the thickness of six grains of barley placed side by side, and 
in sixty trials I have obtained as maximum thickness 17.3 mm., 
us minimum 13 mm., the average of the sixty determinations 
being 15.31 mm. ; which would give us for the length of the cubit 
367.44 mm., a result evidently inexact, by reason of the want of 
delicacy of the standard by which the valuation was made. We 
shall return to this subject later, and shall attempt to find a more 
probable result, such as will show which of the two values is 
nearer the truth. 

In the fifth and last chapter of our author's third lecture, he 
takes up the problem of tne chess-board, of which he supposes 
the squares to be filled with dirhams, each square containing 
twice the number in the preceding. He begins with finding the 

* Exposition da Systeme chi Monde, p. 895, 6m* Edition, 1885. 



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80 N. KhanHcoff, 

total number of dirhams to be 18,446,744,073,709,551,615, ex- 
pressed by him in abujart signs thus : I^OO^cXi^jiL^^lP. 
Then he applies himself to find the dimensions of the treasury 
in which tins treasure should be deposited, and finally cites the 
verses of the poet 'Ansarf, chief of the poets of the Sultan Mah- 
mtid of Ghaznah, which fix the time in which one might spend 
this sum at 200,000,000,000,000,000 years. The verses are as 
follows : 

JLu O )j0J jUj JU }f u-o tj .*. fjfji O };06\ gUt JU^ IpU 

JL* }p vi^xrU •, viiAX:U }f L 5^ . \ j}j Jf> wXao ^U ^ »U ^ iU 

" kins; I live a thousand years in power ; after that, flourish 
a thousand years in pleasure : be each year a thousand months, 
and each month a hundred thousand days, each day a thousand 
hours, and each hour a thousand years." 

Before giving a succinct description of the physical instru- 
ments described and mentioned in the Book of the Balance of 
Wisdom, I think it well to pause and review the results arrived 
at by the Arab physicists, and recorded by our author in the 
first part of his work. I will begin by attempting to give a 
little more precision than has been done hitherto to the units of 
measure, as the cubit and the mithkfil. 

We have seen that the cubic cubit of water weighed by 'Abu-r- 
Raihan at Ghaznah weighed 28,605.647 mithkals. The eleva- 
tion of Ghaznah, according to Vigne, is 7000 English feet, or 
about 2134 metres, which would correspond to a medium baro- 
metric pressure of 582 millimetres. The temperature of the 
water made use of by ; Abu-r-Raihfin in this experiment is not 
known to us ; but not only have we seen our author state in the 
clearest manner that he was aware that temperature had an in- 
fluence upon the density of liquids ; we may also see, upon com- 
paring the specific gravities of liquids obtained by the Arabs 
with those obtained by modern physicists, that their difference 
between the density of cold and of hot water was .041667, 
while, according to the experiments of Hallstrom (see Dove's 
Repert. d. Physik, i. 144-146), the difference between the densi- 
ties of water at 3°.9 and at 100° (Centigrade) is .04044. We can 
assume, then, with great probability, that a physicist so experi- 
enced as 'Abu-r-Rainfin would not have taken water at its maxi- 
mum summer-heat, but that he would have made his experiments 
either in the autumn, as our author advises, or in the spring. 
The temperature of the rivers in those regions in autumn has 
not, to my knowledge, been directly determined by any one, but 
the temperature of the Indus, at 24° N. lat, in February, 1838, 
was measured by Sir A. Burnes (see Burnes' Cabool, p. 307), and 
was found to be, on an average, 64° 2 Fahrenheit, which is 



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Booh of tiie Balance of Wisdom. 



81 



equivalent to 17°.89 Centigrade ; and we may, as it seems to me, 
with sufficient probability, admit that the water used by 'Abu-r- 
Baihan was of a temperature about 62° Fahrenheit, and that its 
density, according toHallstrom, was .999019, considered in refer- 
ence to water at the zero Centigrade, and .998901, considered in 
reference to water at its maximum density. Now we know that 
a cubic metre of distilled water, at 4° C, weighed at Paris in a 
vacuum, weighs 1,000,000 grammes; if, then, we know the value 
of the mithkal in grammes, we shall be able to compare the' 
metre and the cubit. 

According to the K&mds, the mithk&l is If dirhams, the 
dirham 6 dfiniks, the d&nik 2 klrats, the kirat 2 tassfijs; our 
author, however, makes use of a much less complicated mithk&l, 
which is composed of 24 tassfijs ; i£ then, these tassujs are the 
same as those of the KSmfts, his mithkal is equivalent to the 
dirham of the latter. I shall not follow the methods pointed 
out by Oriental authors for determining the weight of the mith- 
k&l, for they are all founded upon the weight of different grains, 
no notice whatever being taken of the hygrometric condition of 
such grains; but I shall pursue an independent method. I 
would remark, in advance, that almost nowhere, not even at 
Baghdad, has the Arab dominion of the first times of the Khalife 
left so profound traces as in the Caucasus, where, as in Daghistfin, 
for instance, while no one speaks Arabic, correspondence is car- 
ried on exclusively in that language. Now I have made, by 
order of the Government, and conjointly with M. Moritz, Di- 
rector of the meteorological observatory at TifliS, a comparison 
of the weights and measures used in the different provinces of 
Transcaucasia, and have found the value of the mithkal in 
grammes to be as follows : 



In the district of Kutais, 


4.776 grammes. ' 


: 


u 


Thelawi, 


4iK7 


M 


► Georgia. 


u 


Sighnakh, 

Nuthjiwan, 

Ordubad, 


4.226 


i 




a 


4.590 


u > 




u 


4.499 


u 




ii 


Shemakhi, 


4.305 


a 




In the city of 


Shemakhi, 


4.704 


u 




u 


« (another), 


4.572 


u 




ii 


a u 


4.621 


a 




In the district of Baku, 


4.175 


it 


"Muslim" 


u 


u u 


4.660 


a 


Provinces. 


li 


Karabagh, 


4.610 


u 




ii 


Sheki, 


4.496 


u 




ii 


" canton Eresh, 


4.272 


u 




a 


« " u (another), 4.426 


a 




a 


" " Khajmasi, 


4.869 


u 




it 


Lenkoran, 


4.660 


a 




u 

(C 


Derbend, 
Samur, 


4.538 
4.792 


u \ 
u 


• Daghistan. 


Average of nineteen values, 


4.527 grammes. 





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82 KKhmikoff; 

Hence it is seen that the particular values vary in either direc- 
tion from this average, to as much as +0.342 gr. and —0.852 gr., 
and I accordingly believe that the value of the mithkfil may be 
taken at 4.6 gr. without fear of any considerable error. 

Accepting, then, 4J grammes as the equivalent of a mithkffl, 
-we shall find that the weight of a cubit cube, 28,605.647 mith- 
k&s, is 128,725.41 grammes. In order to compare with this 
the weight of a cubic metre of water, it will be necessary to 
reduce the latter to the conditions of Ghaznah with respect to 
temperature, atmospheric pressure, and intensity of the force of 
gravity. Calling m * the weight of the cubic metre of water, and 
considering only the temperature of the water of y Abu-r-Raih£n, 
we shall find m*= 998,901 gr., which would be the weight of a 
cubic metre of water in Pans at 16°»67 C. in a vacuum. Now, 
according to the experiments of M. Begnault, a litre of dry air in 
Paris, at zero of temperature, and under a barometric pressure of 
760 mm., weighs 1.298187 gr. ; a metre, then, will weigh, under 
the same conditions, 1298.187 gr. The intensity of gravity at 
Paris, g, is 9.80895 m. ; and at Ghaznah, g 1 , 9.78951 m. ; then d', the 
weight of a cubic metre of dry air at Ghaznah, will equal 981.241 
gr.* As we have no means of ascertaining the hygrometric con- 
dition of the atmosphere during the experiment of 'Abu-r-RaiMn, 
we are compelled to treat it as if perfectly dry : by deducting, 
then, df from m 3 , we shall render this latter number in all respects 

comparable with c 8 , and we shall have — -g =0.1291, and — = 

= 0.505408 : c, then, equals 506.408 mm., a value which dif- 
fers from that which we obtained by comparing the Arab meas- 
urement of a degree with our own, by 14.896 mm., that is to 
say, by about the average thickness of six grains of barley laid 
side by side ; and I think we may assume, without danger of too 
great an error, c = 500mm. Notwithstanding the hypotheses 
which I have been compelled to introduce into this calculation 
in order to render it practicable, the result obtained by it seems 
to me preferable to that derived from a comparison of the dimen- 
sions of the earth, for here we can at least form an approximate 
idea of the amount of possible error, while in the other case we 
are deprived of all power of applying a test, by our ignorance 
respecting the degree of precision of the geodetic instruments of 
the Arabs. 24 
This furnishes us the means of ascertaining the fineness of the 

* The accelerating force of gravity it here calculated by the formulas g f -=-g 
(1 - 0.002588 cos. 21) 1 1 - — V and r = 20,887,538 (1+ 0.001644 cos. 2/) ; where 
fc=S4°, and *=7000 Eng. ft. ; and by the formula d'=d S!K I g ggp~ ) , where 
t is the temperature in degrees of Centigrade. 



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Booh of the Balance of Wisdom. 83 

silver wire by which Abu-r-RaMn measured the side of his 

Abe 
cube, for w = -™r- , or .000924c, or .462 mm. This ne plus ultra 

of the skill of the Arab jewellers will seem to us coarse enough, 
compared with the silver threads obtained by the ingenious pro- 
cess of Wollaston, of which the diameter is only .0008 mm., or 
ruhnr of an English inch. But it should not tie forgotten that 
it is not long since .006 mm* was regarded as the limit of the 
ductility of gold thread, and that accordingly, considering the 
imperfect mechanical means which the Arabs had at their dispo- 
sal, a metallic wire of a thickness less than half a millimetre 
was in fact something remarkable. 

On examining the determinations by the Arabs of specific 
gravities, we see that they had weighed, in all, fifty substances, 
of which nine were metals, ten precious stones, thirteen mate- 
rials of which models were tnaae, and eighteen liquids. The 
smallness of the list ought not to surprise us, for most of the 
substances which figure in our modern lists of specific gravities 
were entirely unknown to the Arabs. What is much more sur- 
prising is the exactness of the results which they obtained ; for 
the coarseness of their means of graduating their instruments, 
and the imperfection at that time of the art of glass-making, ren- 
dered incomparably more difficult then than now this kind of 
investigation, which, in spite of the immense progress of the 
mechanical arts, is still regarded as one of the most delicate 
operations in physical science. It is very remarkable that the 
Muslim physicists, who had detected the influence of heat on 
the density of substances, did not notice its effect upon their vol- 
ume : at least, the dilatation of bodies by heat is nowhere men- 
tioned by our author; and this circumstance, together with their 
ignorance of the differences of atmospheric pressure, introduces 
a certain degree of vagueness into the values which they give 
for specific gravities. In comparing, as I have done in the fol- 
lowing table, our author's valuation of specific gravities with that 
obtained by modern science, I shall regard the former as having 
reference to water at the freezing point, and under a pressure of 
760 mm., both as not knowing what else to do, and as supported 
by these two considerations : first, that we have already noticed 
the slight difference between the densities given by our author for 
cold and hot water, and that which is true of water at the freez- 
ing and boiling points;* secondly, that our author, according to 
his own statement, made the greater part of his determinations 
at Jurjaniyah, which, in my opinion, is no other than the modern 
Kuna-Orghenj, a city situated about four geographical miles from 
the point where the Oxus empties into the Sea of Aral, where 
he was able to raise the temperature of water to 100° C, and 

* See p. 80. 



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84 



K Khanikoff, 



which consequently must be at the level of the sea. The mod- 
ern values of the specific gravities are given, for the most part, 
and when not otherwise noted, from Schubarth's Sammlung 
physikalischer Tabellen (Berlin : 1849) ; a few are taken from 
the Annuaire du Bureau des Longitudes, Paris, for 1853 (marked 
"Ann."), from Schumacher's Jahrbuch for 1840 ("S."), from 
Brande's Encyclopedia ("Br."), from Hallstrom, as cited above 
("H."), and from Gmelin's Chemistry ("G."). The substances 
are arranged in the same order as they have been given in our 
author's tables. 



Substances. 



Specific Gravities : 
; to 'al-Khazim. ace to modern authorities. 



Gold, 


19.05 cast, 


19.258-19.3 


Mercury, 


13.56 


13.557 


Lead, 


11.33 


11.38&-1 1.445 


Silver, 


10.30 


10.428-10.474 


Bronze, 


"■IJSS, 


8.95 Ann. 
a46 « 


Copper, 


8.66 cast, 


a667-8.726||^ 7 A ^ 1L; 


Brass, 


a57 


8.448-8.605 


Iron, 


7.74 forged, 


7.6-7.79 


Tin, 


7.32 English cast, 


7.291 


Celestial Hyacinth, 


3.96 Oriental Sapphire, 4.83 


Red Hyacinth, 


3.85 « Ruby, 


3.99; 4.28 Ann., S. 


Ruby of Badakhshan, 


3.58 




Emerald, 


2.75 


2.678-2.775 


Lapis Lazuli, 


2.69 


2.055; 2.9 Ann. 


Fine Pearl, 


2.60 


u*\«SSF> 


Cornelian, 


2.56 


2.62 


Coral, 


2.56 


2.69 


Onyx and Crystal, 


«~(Onyx, 2.628-2.817 
A0U 1 Mountain-crystal, 2.686-2.88 


Pharaoh's Glass, 


• o 49 j English mirror 
( M flint- 


-glass, 2.45 
" a442 


Clay of Siminjan, 


1.99 Clay, 


1.068-2.63 


Pure Salt, 


2.19 


2.068-2.17 


Saline Earth, 


1.11 




Sandarach, 


.71 


1.05-1.09 


Amber, 


.85 


1.065-1.085 


Enamel, 


3.93 




Pitch, 


1.04 white, 


1.072 


Wax, 


.95 yellow, 


.965 


Ivory, 


1.64 


1.825-1.917 


Black Ebony, 


1.13 


1.18 


Pearl-shell, 


HKSSHSSf"^ 


Bakkam-wood, 


.94 Brazil-wood, 


1.031 


Willow-wood, 


40 


.585 


Sweet Water, 


1. 


1. 


Hot Water, 


.958 boiling, 


.9597 H. 


Ice, 


.965 


.916-.9267 


Sea-water, 


1.041 


1.0286; 1.04 S. 



* At 17°.5 0., according to M. Abich. 



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Booh of the Balance of Wisdom. 85 

Substances. Specific Gravities : 

ace. to 'al-Kh&zini. ace. to modern authorities. 

Water of Indian Melon, 1.016 

Salt Water, 1.134 saturated solution, 1.205 G. 

Water of Cucumber, 1.017 

Water of Common Melon, 1.090 

Wine-vinegar, 1.027 Vinegar, 1.013-1.080 Br. 

Wine, 1.022 various kinds, .992-1.038 

Oil of Sesame, .915 

Olive-oil, .920 .9176-.9192 

Cow's Milk, 1.110 1.02-1.041 

Hen's Egg, 1.035 1.09 

Honey, 1.406 1-450 

Blood of a Man in good health, 1.033 1*053 Br. 

Warm Human Urine, 1.018 1 1 mi 

Cold « « 1.025 J 1#U11 

This table shows us that the Arabs conceived much earlier 
than we the idea of drawing up tables of specific gravities, for 
the first European tables of this character are, according to 
Liber (Hist. Philos. des Progr&s de la Physique, iv. 113-114), 
due to Brisson, who died in 1806. The first person in Europe 
to occupy himself with determining the specific gravity of liquids 
was Athanasius Kircher, who lived 1602-1680 : he attempted 
to attain his purpose by means of the laws of the refraction of 
light. After him, the same subject drew the attention of Galileo, 
Mersennes, Biccioli, the Academicians of Florence, assembled as 
a learned body in 1657, and finally of the celebrated Boyle, born 
1627. The latter determined the specific gravity of mercury 
by two different methods : the first gave as its result 13-^ft", or 
13.76, the other 18££, or 18.857; both are less exact than the 
value found by the Arab physicists of the twelfth century. 

I will conclude this analysis by a brief description of the dif- 
ferent kinds of balance mentioned in this work ; I shall cite the 
text itself but rarely, and only when it contains something 
worthy of special notice. 

Our author first describes a balance which he calls Balance of 
Archimedes, and professes to quote the details respecting its use 
word for word from Menelaus : lip- lip- J^>jJi JslaJI KjIX>. LXP, 
without, however, giving the title of the latter^ work. 

In order to ascertain the relation between the weight of cold 
and that of silver, Archimedes took, according to our author, 
two pieces of the two metals which were of equal weight in air, 
then immersed the scales in water, and produced an equilibrium 
between them by means of the movable weight : the distance of 
this weight from the centre of the beam gave him the number 
required. To find the quantity of gold and of silver contained 
in an alloy of these two metals, he determined the specific grav- 
ity of the alloy, by weighing it first in air and then in water, 
and compared these two weights with the specific gravities of 
gold and of silver. 



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86 



IK Khamkoff, 



This is the figure of the Balance of Archimedes given in my 
manuscript : af 

Figure of the Balance of Archimedea. 




nnhniliflii l ii ii liml 




a. v^^I %& 
Bowl for Gold. 



e. Klkjt 
Movable Weight 



ft. juLalt *£Y 
Bowl for Silver. 



Another balance described by our author is that of Mu- 
hammad Bin Zakariyfi of Bad. It is distinguished from that of 
Archimedes by the introduction of the needle, called by the 
Arabs ^LJJ! > " the tongue," and by the substitution of a movable 
suspended scale for the movable weight. The following is an 
exact copy of the figure representing it :» • 






c. /CoIj &oa)t KaY 
Bowl for Silver, fixed. 



Bowl for Gold, movable. 



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Book of the Balance of Wisdom. 87 

A note inserted in my manuscript between the two bowls, 
wbich I have copied and translated below, explains the mode of 
using this balance. 

o^*x o^T ^ l^ jjo»o iuaJL> &&» *** *^&m f s a tf m 

When the body in question is pure silver, the bowl containing it will 
be balanced at o, which is the extremity of the beam, and the place 
where the scale commences. When it is the purest gold, the bowl will 
come as near to the tongue as possible, at 6. When it is mixed, it will 
stop at A, between a and 6 ; and the relation of the gold in the body 
to what it contains of silver will be as the relation of the parts [of the 
scale] a h to the parts a h ft. Let this, then, be kept in mind with regard 
to the matter. 

A third balance described by our author is that of 'AM- 
Hafs 'Umar Bin 'Ibrahim 'al-Khaiy&nL I do not copy the figure 
of it, because it is in every respect similar to the balance of 
Archimedes, excepting the movable weight. Its application is 
very simple. A piece of gold is weighed in air, and then in 
water ; the same thing is done with a piece of silver ; and a 
piece of metal about which one is doubtful whether it is pure 
gold, or silver, or contains both metals at once, is also tried ; 
and the comparison of specific gravities thus obtained serves to 
settle the question. 

Finally, in the fifth lecture, he gives a verv minute description 
of the balance of wisdom, according to 'Abu-Hfitim 'al-MuzafFar 
Bin 'Isma'il of 'Isfaz&r. He begins by remarking that, the bal- 
ance being an instrument for precision, like astronomical instru- 
ments, such as the astrolabe and the zij 'as-saia'ih,' 7 its whole 
workmanship should be carefully attended to. He next de- 
scribes the beam, *>^H, the front-piece, fccajyJt, the two cheeks, 
"QtjlftftJI, between which the tongue moves, and the tongue itself, 
LJJt. As regards the beam, he advises that it be as long as 
it may, " because length influences the sensibility of the instru- 
ment;" and indicates a length of four bazaar-cubits, or two 
metres, as sufficient He gives to his beam the form of a paral- 
lelopiped, and marks upon its length a division into parts, two 
of which must be equivalent to its breadth. It must be of iron 
or bronze. The tongue has the form of a two-edged blade, one 
cubit in length ; but he observes, as in regard to the beam, 



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88 N. Khanikoff, 

that the longer it is the more sensitive will be the instrument. 
He directs to fasten it to the beam by two screws, after having 
carefully determined the centre of gravity of the beam, by 
placing it, experimentally, across the edge of a knife ; and to 
fit it with nicety, so that the centre of gravity may be as little 
displaced as possible. We do not stop to give the description 
of the tongue and its frame, and limit ourselves to copying 
exactly the accompanying figure, which represents these parts 
.of the balance: 88 




a. xcos^jJt 
Front-piece. 

Bending-place. 

After this, our author exhibits the general principles which 
concern the suspension of the beam of the balance. The passage 
deserves to be transcribed and literally translated, as is done 
below. 

jjs«3 vdttiij ^>A r i£>t j> sjm\ ja\ jjujt ^ 

o* fcJU jib jy^b LJJl ^ L>3L JXiJI ^)j^\ £y+X D tf LM 
J> »& /j* J* pjk ) jfi 3 $J^W ^ 1*^\ «>>3«^ 
D i yJ& U>L^Utt <jJL* ^] D yCo A£> J* Uj15 ^jsuj& *k~> 

Section Fourth. [Led. 5, Chap. 2.] 

Scientific Principles of a General Nature, universally applicable, relative 
to JDetermination of the Axis, the Place of Perforation [for it], and 
the Point of [its] Support [to the Beam]. 

The beam being columnar in shape, detached from the tongue, there 
are three varieties of axis : 1, the axis of equipoise, at the centre of gravity 
of the beam, exactly in the middle of it, and perpendicular to its length ; 
so that the beam readily gyrates in obedience to equiponderanoe [in its 
two equal arms], stopping, in its going round, wherever that [moving 
force] ceases to act, and not becoming, of itself, parallel with. the horizon ; 
because a right line drawn from the centre of the world to its centre of 



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Booh ofOie Balance of Wisdom. 



89 



■ju yiUJf j^o ^* zj^ fi~^ o* £*^ L^t* cJlftSt u^> tot 



jfjA V+ ^jJk ^Jl ***** ^' lib *]# /y> >J&+^\ £M o'^ 

^^tc ffr} *i* &4-~& o-^^ ui^^ {j&*&s*u&&jf jA ^\ j\jA\ 

IgJuaJ* *♦*»&> UPlP ^£ iW J1 ^ vj&jt a! JLs^ Jot uAA^ £*?"^ f^ft? 



V ' U" 


« t 


a 




-> 


r 




o 


» 8 




m 


s 




'a 0. 

* 


4 



gravity cuts it into like halves wherever it stops; 2, the axis of reversion, 
between the centre of the world and the centre of gravity of the beam, so 
that, when the beam is pat in motion, it turns, of itself, upside down, be- 
cause a right line drawn from the centre of the world [through the axis, 
when the centre of gravity is thrown out of that line] divides it into two 
parts differing one from the other, of which the one going downward pre- 
ponderates, and the beam is consequently reversed ; 3, the axis of [paral- 
lelism by] necessary consequence, above the centre of gravity of the beam, 
so that, when the beam is put in motion, a right line drawn from the 
centre of the world to its centre of suspension divides it into two parts 
differing one from the other, of which the one going upward exceeds 
in mass, and consequently preponderates and returns, and so the beam 
•tops in a horizontal position ; because, in this case, a right line [drawn 
from the centre of the world to the point of suspension] divides it 

12 



VOL. VI. 



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90 N. Khanikoff, 

»LU> tili w^jJl v)^ j*V° Ck^-^ t5^* 8 fclaJUj ^i* &&aou Qj*} 
u£ <j* ^s?. L 5«AJI ff*"Jt n£»(j* Q*i J^gJ vi^A> \J&j le\ya* \^y^ 
^%J^Ly«JC8 ^v-^aJb v>^v-j! ^la^ f-"**M J^t j^/* » ^' ^Ltll j*j* 
jO° Cf * vJL^ UU> Wj *£> ^i JL tit t«A0> a^s* JjLj „£j 
q>£jj tj^&Lcs^ [;£+*&* ^JauJt ^♦JwJij (j*^w^ Jai£ *£>■ ^t ^JJ&lt 
b o^' ^ Lite- tJt^ vJii^Jt ^Bt^tyj j^Ls £>^ g^J** f^' J*J**N 
| ^Ju ^^Ji "i g^t JjUI yCjO X^> ^t JU, JJ&t j5y> ^ 
^3^3 q* vJ&Lo f*z>$ O-^* *^*W * kfi*% ^AjalUi^ ^^vwib ^vj' 
UUi Jjtf *Utf J>\ j^aSi tit W, ^ g 3LJI f£>^ twX^ v^' ^1 

into two like halves, and parallelism with the horizon is a necessary 
consequence. 

Let abjd 29 be a detached beam, let the line inn divide it into halves, 
lengthwise, and the line «'a halve it across, and let h, the point where 
the two lines meet, be the centre of gravity of the beam. When, there- 
fore, we set the beam on an axis [at that point], so that it obeys [the 
equal weights of its two arms], it stops wherever it is left to itself; 
because the right lino shk, drawn from £, the centre of the world, to A, 
the centre of gravity, divides the plane abjd into like halves, according 
to an explanation which it would take long to state. This equal divis- 
ion occurs, however the beam may incline. When we set [the beam 
on an axis at r,] above A, away from the centre of gravity, then the line 
krs [drawn from the centre of the world to the point of suspension] 
divides the plane into two parts differing one from the other, of which 
the one going upward has the greater bulk, so that it preponderates 
and returns, and parallelism with the horizon is a necessary consequence. 
When we set the beam on an axis at h, below A, away from the centre 
of gravity, and the beam leans, then that part of it which goes down- 
ward preponderates; because the right line [drawn from the centre of 
the world to the point of suspension] divides abjd into two parts differ- 
ing one from the other, and the mass 30 going downward preponderates, 
so that the beam turns itself upside down. 

So much for the beam when detached from the tongue. 

In case of the combination of its own gravity with the gravity of the 
tongue, placed at right angles to it, in the middle of it, the common cen- 
tre of gravity differs from that of the detached beam, and must neces- 
sarily be another point ; and that other point corresponds to the centre 
of equipoise in the detached beam, so that, when the beam is set upon 
an axis [at that point], it stops wherever it is left to itself. 



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Booh of die Balance of Wisdom. 



91 



\jy£ J*> 131 Juli S 3LJI £ Jl Jtffifl j^ ^ *£> yWB (dJj^ y£f 



V6 / 


* 


a 




J 


J 




O 


n s 


h f 

5 


77! 


s 


e 
4 


/ ' 

'a 


rf 



JuUJ^ c^***** gJa*Jt ^*"* A 4*N zv)^ ffr*Jt o^ (*IjaW! j L ^ 

The tongue may be made fast above [the beam], in the direction of 
*, the point I becoming the common centre of gravity ; and an axis at 
this point is the axis of equipoise. So that the axis of [parallelism by] 
necessary consequence is at any point fixed upon above /, because a 



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92 K Khanxkoff, 

v5 **& <&Joj> fcJaiu Ji^ vjiitf? H^t^o ^c v-aSj^ £^"jA* £^;' CL^ 
t ^t J**-* ^bct aJwo ±Al\ r &\ a yu JU U> vttftftt ^^ 
xUj* jS'wo q>^a» p _>^ ^+*^ vi>^ # qLJJ5 {joji 131^ Lm^ow v-U*^ 
i>^J J**> kite J^-gJ vi^> s-ARJ *&*>- !^>li JliAXcyi i>^ >P LK> *k& 
131} vj&ft aloLs? Jj: v—AJM^ Jj.L^JJ ff^J*? f\y$i\ \y^ jf^tp ijo yJLy* 

K^> q* cju^ v-j L^JiLo^ U^Xmiuo qLmJLS! ** i-*^-^ L>3Lm &S^ a^> 

*JL> Lo^ Lfc^j ^q^Xcj mum tfcXp xX^'^t &3y» ^t «^ aUt^ Jo: 4JL0 
JjjuXjp* oU^IS aA^J LuLot iASj LgJLo **^ Jjff 

right line drawn to that [higher point, from the centre of the world] 
divides the plane [abjd] into two parts, of which the one going upward 
preponderates, so that it returns, and stops in a horizontal position; 
and [an axis] at any point fixed upon below / is the axis of reversion, 
so that, when the beam inclines, that part of it which goes down- 
ward has the greater bulk, and consequently tips until it turns upside 
down. 

Should the tongue be made fast below the beam, in the direction 
of 'a, and so the common centre of gravity become the point f, then 
[an axis at this point] is the axis of equipoise, and, therefore, when 
the beam is put in motion, it stops wherever it is left to itself. But, 
when the axis is put above s, it becomes the axis of [parallelism by] 
necessary consequence, so that the part [of the beam] which goes upward 
returns, and stops in a horizontal position. When the axis is put below 
s, it becomes the axis of reversion. 

Inasmuch as there is change [in the adjustment of the balance] in 
several ways : 1, in respect to the beam's being either detached or joined 
to the tongue — [the tongue] standing up or reversed [according as it is 
made fast above or below the beam] ; 2, in respect to the [position of the] 
axis [in each of the cases supposed with reference to the connection of 
the beam] either at, or above, or below, the centre of gravity; '3, in 
respect to the place on the beam of the means of suspension of the two 
bowls, either even with, or above, or below, the axis — twenty-seven 
incidents [constituting changes of adjustment] are made out, together 
with a result [as regards the action of the balance] dependent upon 
each particular adjustment, and we have drawn up for these incidents 
the following table : 



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Booh of the Balance of Wisdom. 



93 



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rfJ 3 * 



um\ 



A* 



JiJOef 



^ISut 



j^Uu! 



f tjXfl jtoael 



wUSit 



ft/R 



fljdl 



v^t 



r |jXtt JbttJ 



j*UI 



^Lail 



ftjdl 



JU*«I 



V*"t 



<¥» 



f»>« 



V^iil 



f <jSj| IjtJOe! 



wbUu! 



V^i! 



fljril 



JtOOel 



V*»M 



^JOst 



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V^iii 



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Sc 



it 



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ri 









NC: 



ft* 



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94 



N. Khamkoff, 



Table of Variety of Incident pertaining to the Balance.* 1 






Necessary ParaUelism, 



Necessary Parallelism, 



Reversion. 



Seek*** 



Reversion. 



Reversion. 



Necessary 
Parallelism. 



Equipoise, 



Reversion. 



Necessary Parallelism. 



Necessary Parallelism, 



Reversion. 



■ Necessary 
Parallelism. 



Equipoise. 



Necessary Parallelism, 



Necessary Parallelism. 



Reversion. 



Necessary | w . . 
Pa I alleli8kH uipo,8e ' 



Reversion. 



Reversion. 



Necessary 
Parallelism. 



Equipoise, 



Reversion. 



Reversion. 



Reversion. 



Necessary 
Parallelism, 



Equipoise, 



Reversion. 




Equipoise. 



Necessary 
Parallelism, 



Reversion. 



Equipoise. 



Necessary 
Parallelism, 



Reversion. 



fin K 

n 



9 



liHl 






'-Ml 

i-g.il 






ill 



f 



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Booh of the Balance of Wisdom. 95 

The author farther describes the bowls of the balance, five 
in number. He advises to make them of very thin plates of 
bronze, and to give to three of them the form of hemispheres, 
measuring thirty divisions of the scale of the beam in diameter. 
The bowl destined to be plunged into water was finished at the 
bottom with a cone, in order that it might more easily overcome 
the resistance of the fluid during the immersion. The remaining 
bowl was spherical in shape. I here give the passage describing 
this last bowl, remarking only that it will be found not to corre- 
spond altogether with the figure d in the representation of all 
the bowls together, presently to bfe introduced, although evi- 
dently referring to that one. 33 

*jc\iL Jaj^ ^! v^^Ia&jt y& lPj> ^J^ K*mJ> ^vXSj e£r*^ { y l^j^j 
*k&5 »y L$OJt vJaftJi ^ uXjlJI oJ^ ^fC -T^ J -b L?Jo* 

Syjb L^jJLcx y.^ xib mL^uaSI ^^ ^^JLc k^** Jc>li ^ jLoJi cr 
>S ^'j_5 Lo LgJU v-iA^ij jLiii! q* Pi> y£*& **mj> ^Pj L>o ^o&> 

Then we take a fourth clepsydra, [turning] on an axis A, of which the 
diameter measures thirty divisions [of the scale of the beam], as does that 
of the two air-bowls ; and we cut it on the two sides [of the axis], meas- 
uring five divisions [once and twice] in the direction of the axis, towards 
the centre of the bulge — of which cuts one is tnl and the other hmk — 
leaving, between the axis h and the point n, a distance of five divisions, 
and, between k and the point m, a distance of ten divisions, and calling 
tl the inner side, and hk the outer side; so that the remainder [of the 
diameter of the clepsydra] between the two cuts measures fifteen divisions 
of the standard-measure. In the next place, we take a thin plate [of 
bronze], as large as the clepsydra, and mark upon it a circle opening with 
a certain spread, namely, of fifteen divisions of the standard-measure, and 
cut off from that [plate] all that is outside of that [marked circle] ; after 
which we cut that [circle] into two unequal parts, bend each part, and 
weld it, separately, to one of the outer edges of the two sides ; and we 
call this the winged bowl. 

Two of these bowls bore the name of " the aerial," A&ri^l f 
and were permanently attached to the beam. Another bowl, 



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96 



2T. Khxmikoff, 



hemispherical in shape, might be moved along w the right arm of 
the beam ; it was called "the movable" bowl, Xliult . The spher- 
ical bowl, also, was moved along the right arm. The bowl in- 
tended to be plunged into water was made fast underneath the 
aerial bowl of the left arm, and bore the name of " the aquatic" 
bowl, &5.UR. As to the §pherical bowl, its name, sufficiently 
explained by its form, just now described, and given to it in the 
extract, was "the winged" bowl, X^U^it. Our author adds 
that it was indispensable to have at least one movable bowl, 
in order to balance the tw# which were used when the body 
weighed was plunged into the water. 

The following is a copy of our author's drawing of these five 
bowls grouped together : 




a. i5UJI J£\ WlR 
First, Right-hand Bowl. 

b. L $y~3\ &ol*J! XAift 
Second, Left-hand Bowl. 

c. Itl Jlfc XU^I KSJUI! S*JSt 

it*. 

Third, Conical Bowl, called the 
Judge. 



Fourth, Winged Bowl, cut on the 
two sides. 

Inner, Nearer Side. 

Fifth Bowl, which is the Movable 
Bowl. 



Having devoted a paragraph to describing the form which 
should be given to the rings of suspension for the bowls, all of 
which are shaped like m in the figure on the next page, the 
author at length presents a complete drawing of the balance of 
wisdom. This we here reproduce, with all the accompanying 
explanations : 3 s 



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Book of the Balance of Wisdom. 

£*LiL u^yuJl *£& G ^ t^yo 
Figure of the Balance of Wisdom, called the Comprehensive. 



97 




above, on the left : above, on the right : 

f>)y?& y~Si\ vJUuJi Left Half; oL?Uao1I ^\ s^aJI Right Half, 
for Substances. for Counterpoises. 

along the beam, on the left : along the beam, on the right : 

H-^LfcJt ot-ui£Jt Plain Round- iuilii ot-oi£Jt Hidden 8 * Round- 
point Numbers. point Numbers, 
under the beam, on the right : 
wJlil kXP ^ julfi J^y oJyj; The Specific Gravities are marked on 

this Side of the Beam. 
a. y^^al! Means of Suspension. /. *JJj\j$\ ^jtJl Air-bowl for the 

End. 



b. JuajyJ? Front-piece. 

c qL**1N Tongue. 

d. " l »;lftAJI Two Cheeks [of 

the Front-piece]. 
«, and under the beam on the left: 

Jutfo^'The Front-piece and the 
Tongue as disconnected [from the 

VOL. VI. 13 



g. X-oUJ? A-u|^Ji KaI^JI Second 
Air-bowl for the End. 

h. itfJUJI *UI Ktf Third [or] Wa- 
ter-bowl. 

t. xjutyti^U^I Fourth [or] Wing- 
ed Bowl. 

j. X^cliiiJJUl! Fifth [or] Movable 
BowL 



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Ni Khanticoffi 

con- k. JJL* 

- aent J. JJJL.w. „~. W1 ~~,. -, 

with regard to the Place of Con- z7' A ^ > " . , .\ 

nection * Pomegranate-counterpoise, which 



Beam], because they are not con- jfc. JJLaJf Basin. 

nected until after Experiment j y ,1 ^JtUUuH^jja V LuJj mU Jt 

with regard to tha Pla^fi of Hon- rP ^ > ••• . _ .•*_ 



rides upon the Rising Half. 

Our author recapitulates, briefly, his description of the different 
parts of the balance of wisdom, and then proceeds to speak, in 
detail, of the mode of adjusting it. Nothing of what he says on 
this point deserves to be cited. I will only borrow from it the 
observation that the Arab physicists were accustomed to mark 
the specific gravities of different bodies, on the right arm of the 
beam, by points of silver enchased at different places along the 
scale, where the movable bowl was to be put in order to coun- 
terbalance the loss of weight of different metals and precious 
stones when plunged into water. This accounts for the term 
o|^*-iJt, "round points," applied to marks of specific gravity 
upon the beam of the balance ; and similar usage in respect to 
all marks of weight upon the beam led to the more general ap- 
plication of this term. 

But, before proceeding to describe the application of this bal- 
ance to the examination of metals and precious stones, as to their 
purity — which will bring out all the workings of the instrument 
— I think it incumbent upon me to transcribe and translate the 
following passage, which is, without doubt, one of the most re- 
markable in the whole work : 

JUS &u£>2 ,£& 

JLjlJj iuuUJi auo; OvGjij J5JW uXJbj &aaa && pU sL+Jt Q* 1X^3 

Section Fifth. [Leek 5, Chap. 4.] 
Instruction relative to the Application. 

Air- weight does not apparently vary, although there is actual varia- 
tion, owing to difference of atmospheres. 

As regards its water-weight, a body visibly changes, according to tfie 
difference between waters of [different] regions, wells, and reservoirs, in 
respect to rarity and density, together with the incidental difference due to 
the variety of seasons and uses. So then, the water of some determined 
region and known city is selected, and we observe upon the water- 
weight of the body, noting exactly what it is, relatively to the weight of 



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Book of the Balance of Wisdom. 99 

kL>. v^Ut D5 v> JUJI *Ul *Lc£J! j J^j Q ! UA4P ^Iju *IR ^U Q l 

one hundred mithk&ls ; and we refer [all] operations to that [result, as 
a standard], and keep it in mind against the time when we are called 
upon to perform them, if the Supreme God so wills. 

In winter, one must operate with tepid, not very cold, water, on 
account of the inspissation and opposition to gravity of the latter, in 
consequence of which the water-weight of the body [weighed in it] 
comes out less than it is found to be in summer. This is the reason 
why the water-bowl settles down when the water has just the right 
degree of coldness, and is in slow motion, while, in case it is hot and 
moving quickly, or of a lower temperature, yet warmer than it should 
be, the bowl does not settle down as when the water is tepid. Tfie 
temperature of water Js plainly indicated, both in winter and summer; 
let these particulars, therefore, be kept in mind. 

'Abu-r-Raihan — to whom may God be merciful ! — made his observa- 
tions on the water-weight of metals and precious stones in Jurjaniyah [a 
city] of Khuwarazm, early in autumn, and with waters of middling cold- 
ness, and set them down in his treatise already spoken of. 

This passage puts it beyond doubt that the Muslim natural 
philosophers of the twelfth century knew the air to have weight, 
though thev were without the means of measuring it. The 
sentence italicized would lead one to believe that they had some 
means of measuring the temperature of water ; and, not to resort 
to the supposition that they possessed any thermometrical in- 
strument, even of the sort used by Otto Guericke, which was 
a balance, I think that they simply used the areometer for that 
purpose; and that this instrument was the means of their 
recognizing that the density of water is greater exactly in the ratio 
of its increase in coldness. 

As a last citation of the words of the author whose work we 
have been analyzing, I shall transcribe and translate the passage 
in which he exhibits the application of the balance of wisdom to 
the examination of metals and precious stones, with regard to 
their purity. It is as follows : 



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100 N. KhanOoff, 

ijc^ \ Li ^JJ* iu^/y oL% l^i L^Co? o* \&jk Otf s% 

^vxlLj^ jJ>« h> cr ^!>^ ^j^t uJL> o 1 *^ ** ^ ! L5 5 
or > C 5vjU 5 b^U *4jL^ *0>Ia£!j *U*^ q* 0O5 v^!» <^" o-^ 

vJy ^ *UI j xjut> ysAalrtit |^JI ^ U^ *J«J 3 U> v->JS* 

O^SJ ^ ^U«JI 3 X3U.SI, J*i* JmJI "MfSlJ* '^ 

jUS! yy ^ Xi&JI £to> iAju objiJI q* uVj>! 3 iXs*^ G L^ot ^ 

Chapter Fourth. [Lect 6.] 

Application of ike Comprehensive Balance. 

Having finished experimenting with the balance, and fixing upon it 
the [points indicating] specific gravities, it only remains for us to go 
into the application of it, and the trial of a [supposed] pure metal or 
precious stone, by means of the two movable bowls [tnat called u the 
movable" and " the winced"], reference being had to specific gravity, 
with the least trouble and in the shortest time, by way of distinguishing 
[such metal or precious stone] from one which is alloyed, or from imita- 
tions, or from its like in color — the substance being either simple or 
binary, not trinal, nor yet more complex. 

We adjust the two air-bowls of the balance, put the water-bowl into 
the water, and then set the movable bowl at the [point indicating the] 
specific gravity of the given substance, and equilibrate by means of the 
pomegranate-counterpoise and the scale, until the tongue of the balance 
stands erect. Thus we proceed when the trial respects simple sub- 
stances. When the trial is in reference to a mixture of two substan- 
ces, or a fancy-likeness in color, we set the two movable bowls at the 
Etwo points indicating the] specific gravities of the two substances, and 
►ring the balance to an equilibrium, with the utmost precision possible, 
and make the trial. 



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Booh of the Balance of Wisdom. . 101 

££*> ^\ sUi\ JLojI £ J?Lto^l y 2UJUM ^ U (i)J3 j ob^' '■H*^ 

**U Utj UaJL>^ {J *JS jUH Ut 3 l^ D tf \ jpjp jj^U wJ^ i* 
VuLi G tf t^ Jfcp^B r/ ^L iai*>l5 J^LSUlf vil4 4^AB a K l* ^ 
*!>* 3^ ^-b^ 1 »bl*o, juj^ &*& a^ Xyio >i to! Uj *JU JJtfSL* jliH 

oljUif ^jWt «/«JM5 a*^/*^ J^- cr» «&* **** ^ ^ v^% 

Section Firot. 

2VwZ of Single Simple Substances, after placing the Movable Bowl at 
the [Point indicating the] Specific Gravity of the Metal [or Precious 
Stone], and after the Poising of the Balance. 

When that is the trial which we wish to make, we weigh the substance — 
it being on the left, and the mithkals on the right, in the two air-bowls ; 
then we let it down into the water-bowl, until it is submerged, and the 
water reaches all sides and penetrates all parts of it If there happens 
to be a perforation or a hollow place in it* that must be filled with 
water ; and the weigher endeavors to have it so, taking all possible care 
that the water reaches all its parts, in order that there may remain in it 
no hollow place, nor perforation, containing air, which the water does 
Dot penetrate, because a void place in the substance has the same effect 
as if it were mingled with something lighter than itself. After this 
we transfer the mithkals from the extreme bowl [on the right] to 
the movable bowl, placed at the [point indicating the] specific gravity 
of the substance ; whereupon, if the balance is poised, and stands even, 
not inclining any way, the substance is what it is [supposed to be], pure, 
whether a metal or a precious stone. Should the balance lean any way, 
the substance is not what it is [supposed to be], if a precious stone ; 
and, as to the case of a metal, it is not purely that, but only something 
like it, different from it. If the rising [of the beam] is on the side 
of the mithkals, the substance [being a metal] is mixed with some 
body heavier than itself ; if on the side of the substance, then with 
some lighter body. 

On the other hand, since the substance may not be an imitation, but 
may have been tampered with, and expressly made hollow, blown with 
air, fissured, or the like, trickishly, let that be looked out for, and made 
manifest^ w ^ regard to metals, by striking them. 



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102 • K. Khanikoff, 

Section Second. 

Trial of a Binary, made up of any two Substances [supposed], e. g. of 
two Metals, and similar to Gold; whereby the Assignment of their True 
Value to JDirhams and Dindrs is determined. 

After having adjusted the two extreme bowls and the water-bowl, we 
set the two movable bowls at the two [points indicating the] specific 
gravities of the two metals supposed, or, one of them at the [point indi- 
cating the] specific gravity of a precious stone, and the other at [the 
point indicating the specific gravity of] its like in color, crystal or 
glass ; and then we poise the balance, with the utmost exactness, until 
its tongue stands erect. Then we weigh the body [under examination] 
with the two air-bowls, taking the greatest care ; and in the next place 
dip it into the water-bowl, being careful that the water reaches all its 
parts — which is a matter that the weigher can manage, as it respects 
void places in sight, or seams, so as to be able to remove uncertainty — 
after which we transfer the mithkals [from the air-bowl on the right 
hand] to the movable bowl suspended at the [point indicating the] 
specific gravity [of one of the two substances supposed], and watch the 
balance. If the balance is in equilibrium, the body is that substance, 
pure. If it is not even, we transfer the mithkals to the other movable 
bowl ; and if the tongue then stands erect, the body is a colored imita- 
tion, having naturally the [latter] specific gravity. These remarks apply 
especially, though not exclusively, to precious stones. 

In the case of metals, when neither movable bowl brings the bal- 
ance to an equilibrum, the body is compounded of the two [metals 
supposed] ; and, if we wish to distinguish [the quantity of] each compo- 



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Booh of the Balance of Wisdom. 103 

^^aj ^UJ! Lej 3 lib L$JU tXs><3 J*' ^y> Jr>>y,<vLt ^ Lt j+*£ li^t 

*i^«tf juui \jJi\ Jjoju D t A! Ub l y**?. )*&, "^l A» (^OuuJI 

*jL* v>J *&U ^jJU J-^l bj^ct I3l> jCJt J^t ^ J*3l£Ji Xjj 

Dent of the mixture, we distribute the mithkals, at once, between the 
two movable bowls [suspended at the two points indicating the specific 
gravities of the two metals supposed], giving some of them to the 
movable bowl [so called], and some of them to the winged bowl, and 
watch. If, then, the right side [of the beam] goes up, we transfer [mith- 
kals] from the bowl nearer to the tongue to that which is farther from it ; 
and, if the right side goes down, we transfer mithkals from the farther 
bowl to the nearer; and so on, until the balance is in equilibrium. Then, 
after it is even, we look to see how many mithkals are in the bowl [sus- 
pended at the point] of the specific gravity of a [supposed] metal, and 
those constitute the weight of that metal in the compound ; and the 
mithkals in the other bowl constitute the weight of the other com- 
ponent If we fail to distribute exactly between the two bowls by mith- 
kals, we take the weight of the mithkals in Makkah-sand, or, when sand 
is not to be had, sifted seeds supply its place ; and we distribute the sand 
[or seeds] between the two bowls. When the balance is brought to an 
equilibrium, we weigh what is in each of the two movable bowls, and so 
is obtained a result as perfect as can be. 

Should the balance not be made even in either the first or the second 
instance [namely, by putting all the mithkals in one or the other of the 
two movable bowls], nor by distribution [between the two bowls], then 
the compound either does not consist of the two substances which may 
have been mentioned, or is composed of three or more substances ; or . 
else the two [as compounded together] have been tampered with, and pur-^ 
posely fissured or hollowed. A cavity gives occasion for transfer of 
gravity and weight One must be careful and considerate, therefore ; and 



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104 N. KhanUccff, 

J& v-Jlil JJtf> La>t JU ^^t ^1 JJiiJt JJUj (jvoli^ <^>5 JLa 
JU> Jju Soli i^-olii Jot JL& !3t L«5j *4j4*Ji ^ »l^J ^ UuL-g 
j/ 3 ^- o' v*** **°.* £>*> o' v^a* L*** 9 v*^r* ^p r*^ v^ 

v^A^pvJf V s ***? *3iU?, q^» OiiL) XrfaftJf 

the way to be considerate is to watch [the balance]. If; bow, one of the 
two sides [of the beam] goes up, and if, upon the transfer of gravity to 
the other movable bowl, this side still goes up, the tampering which we 
have spoken of is made certain. If one of the two sides goes up, and 
then, when there has been a transfer [of gravity], the other side goes 
up, the body is compounded of the two substances. 

The distribution [of weight] must be made agreeably to instructions ; 
and one must beware of being deceived in the second case concerning 
it> for example, in the case of a compound of gold and silver [supposed, 
but not proved by distribution] ; and, considering that there may be 
some hollow place within, which opposes [the discovery in it of] gold, 
and makes it [appear as if] of the lightness of silver, one should remove 
its weight to the bowl [adjusted] for silver ; whereupon, by reason of a 
hollow place, one's conclusion may be changed. 

It is evident from this passage that the Muslim natural philo- 
sophers of the twelfth century had so elaborated the balance as 
to make it indicate, not only the absolute and the specific gravity 
of bodies, but also, for bodies made up of two simple substances, 
a quantity dependent on the absolute and the specific gravity, 
which may be expressed by the formula 

1 1 

Trr d r 8. gr. 
x=W— ^-> 

where IT is the absolute weight of the body examined, s. gr. its 
specific gravity, d', d" the densities of its two supposed compo- 
nents, and x the absolute weight of the latter component. 8 * In 
order to accomplish that object, however, they were led to make 
their balance of enormous dimensions, such as rendered it very 

Vjnconvenient for general researches. 
A I will bring this analysis to a close by a concise exposition of 

~j^jfe manner in which the Muslim natural philosophers applied the 
a \nce to levelling and to the measuring of time. 



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Book of the Balance of Wisdom. 105 

The balance-level consisted of a long lever, to the two ends of 
which were attached two fine silken cords, turning on an axis 
fixed at a point a little above its centre of gravity, and suspended 
between two sight-pieces of wood, ws^^JI, graduated. At the 
moment when the lever became horizontal, the cords were drawn 
in a horizontal direction, without deranging its equilibrum, and 
the divisions of the scales of the sight-pieces, corresponding to 
the points where the cords touched tnem, were noted. For lev- 
elling plane surfaces, use was made of a pyramid with an equi- 
lateral, triangular base, and hollow and open to the light, from 
the summit of which hung a thread ending with a heavy point 
The base of the pyramid thus arranged was applied to the plane 
which was to be levelled, and carried over this plane in all 
directions. Wherever the plane ceased to be horizontal, the 
point deviated from the centre of the base. 

The balance-clock consisted of a long lever suspended similarly 
to the balance-level. To one of its arms was attached a reservoir 
of water, which, by means of a small hole perforated on the 
bottom of it, emptied itself in twenty -four hours. This reservoir, 
being filled with water, was poised by weights attached to the 
other arm of the lever, and, in proportion as the water flowed 
from it, the arm bearing it was lifted, the weights on the other 
arm slid down, and by their distance from the centre of sus- 
pension indicated the time which had elapsed. 

[Recapitulating, now, briefly, the results brought out in this 
analysis, we see : 

1. That the Muslim natural philosophers of the twelfth century 
were much in advance of the ancients as regards their ideas of 
attraction. It is true, they ventured not to consider this attrac- 
tion as a universal force ; they attributed to it a direction towards 
the centre of the earth, as the centre of the universe; and they 
excluded the heavenly bodies from its influence.* • Yet they 
knew that it acts in a ratio of distance from the centre of attrac- 
tion. As to their strange supposition that the action of this 
force is in the direct ratio of the distance, having gone so far as 
they had in physics, they must very soon have discovered that 
it was not in accordance with nature. 

2. That they had sufficiently correct ideas respecting certain 
mechanical principles ; that they knew the equation which con- 
nects velocity witn space traversed and time employed in going 
over it; that they were in possession of several theorems relative 
to centres of gravity ; and that the theory of the loaded lever was 
very familiar to them. 

8. That, without yet daring to reject the ideas which had been 
handed down to them by antiquity as to heaviness and lightness, 

VOL. vi. 14 



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106 N. Khanikcff, 

they already recognized that the air has weight, by the influence 
which it exerts upon the weight of bodies. 

4. That they ooserved the action of a capillary force holding 
liquids in suspension within tubes of small diameters, open at 
both ends. 

5. That they made frequent use of the areometer, which they 
had inherited from antiquity, and that this instrument, very 
probably, served them for a thermometer, to distinguish, by 
difference of density, the different temperatures of liquids. 

6. That they already had sufficiently full and accurate tables 
of the specific gravities of most of the solids and liquids known 
to them. 

7. That they had attained, as Baron v. Humboldt very cor- 
rectly remarks^ to experimentation ; that they recognized even 
in a force so general as gravity, acting upon all the molecules of 
bodies, a power of revealing to us the hidden qualities of those 
bodies, as effective as chemical analysis, and that weight is a key 
to very many secrets of nature ; that they formed learned asso- 
ciations, like the Florentine Academy; and that the researches 
of the students of nature in Khuwarazm, of the twelfth century, 
well deserve to be searched for and published. 

Here an inquiry very naturally suggests itself. It is generally 
known that, at the time when the taste for arts and sciences 
awoke to so brilliant a life in Europe, the Arabs powerfully 
influenced the development of several of the sciences. How 
comes it, then, that their progress in physics can have remained 
so completely unknown to the learned of Europe? The answer 
seems to me perfectly simple. The immense extent of the Kha- 
ltfate was a cause which produced and perpetuated the separation 
and isolation of the interests of the various heterogeneous parts 
which composed it A philosopher of Maghrib would doubtless 
understand the writings of a philosopher of Ghaznah ; but how 
should he know that such a person existed ? The journeys so 
often undertaken by the Arabs were insufficient to establish a 
free interchange of ideas ; even the pilgrimage to Makkah, which 
brought together every year representatives of all the nations 
subject, whether willingly or unwillingly, to the law of the Mus- 
lim Prophet, failed, by reason of its exclusive character, to modify 
in any degree that separation of moral interests which kept the 
different Muslim countries apart from one another. Moreover, 
the crusades had an effect to intercept communication between 
the Muslim East and West At length, the Mongol and Turkish 
invasions split the Muslim world into two parts wholly estranged 
from one another, and, so to speak, shut up thejscientinc treasures 
of each part within the countries where they were produced. If, 
now, we reflect that the era of the renaissance in Europe pre- 
cisely coincides with that same invasion of the Turks, we snail 



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Book of the Balance of Wisdom. 107 

clearly see why Europe could scarcely at all profit by the 
scientific monuments of the East of the Khalifate, and why the 
scientific experience of true Orientals has been almost entirely 
withdrawn from its notice. 

Let me be allowed, in conclusion, to add a single observation, 
which is, that it is an error to attribute to Arab genius all the 
great results that the East has attained in the sciences. This 
error rests upon the fact that most of the scientific treatises of 
Orientals are written in the Arabic language. But would lan- 
guage alone authorize us to give the name of Boman to Coper- 
nicus, Kepler, and Newton, to the prejudice of the glory of those 
nations which gave them birth ? Should, then, 'al-Hamadant, 
'al-Flruzabadf, 'al KhaiyamJ, and many others, figure in the his- 
tory of science as Arabs, only because they enriched the litera- 
ture of this people with the Makamat, the Kamus, expositions 
of the KurSn, pnysical researches, and algebraic treatises ? It 
would be more just, as it seems to me, to restore these to the 
Iranian race, and to suppress the injuriously restrictive name of 
Arab civilization, substituting for it that of the contribution of 
the Orient to the civilization of humanity. 

9 Nov. 



Notes by the Committee of Publication. 

Besides re-translating the Arabic extracts in the foregoing article, 
and making other changes which are specified in the following notes, 
we have freely altered whatever seemed to us to admit of improvement, 
being desirous to do full justice to so valuable a communication, accord- 
ing to our best judgment and that of scientific friends who have aided 
us, and fully believing that our correspondent, if we could have con- 
sulted him, would have approved of every alteration which we have 
made. Comm. of Publ. 



I. Notes on the Text. 
Referred to by Letters. 

4. L 8, a, ms. X*yF ; 1. 11, ft, ms. X-yoa3. 
6. 1. 3, c, ms. lAXsA>. 
8. 1. 11, €f, ^yft omitted in ms. 

11 1. 6, e, ms. *&j\juM ; 1. 9, r, ms. XfcJo ilj ; 1. 14, g, ms. L*li/6, 
A, ms. IfJi 



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108 Committee of Publication, 

F*gt. 

12. 1. 8, «, me. u^L&U^; 1. », J, ma. ^V*; L 18, *, ma. a; W f 
I, ma. Q^iil. 

15. 1. 6, an, ma. ^ ; L 14, n, ma. ^j^JlhM , and so wherever else the 
name occurs. 

IA l.l,# ) ma.lf«^«; L4,p,ma.y^; l.ll^^^ycNJt omittexlinWa. 

16. L 13, t% ms. v^t* 

17. L 2 f #, ma. ***; 1. 7, I, ma. **«. 

18. 1. 6, tf, ma. ma>LJ^ ; 1. 8, «, ma, jjryatl— see note 8, p. Ill; 
18. 1. 8, tr, w|^l a*m3- -omitted in ma. 

80. 1. 11, #, ma. (j*9^t*« 

8L 1. 2, t/, vW ***/ omitted in ma. ; 1. 12, jr, | omitted in ma, and 

ao wherever elae this numeral appears in the table of contents. 
88. L 3, « 2 , ma. vi^JI ; 1. 9, A*, ma. I443JU ; L 11, c*, ma. t^gjfN. 

83. L 5, 4i 2 , ma. Js*»Ui — Thia correction ia required by the statement 
of the contenta of the second and third parte of the work gfopn 
on page 17; L 11, e a , ma. a«**i> — aee preceding note. The num- 
bering of the chapters of thia lecture haa been altered in accord- 
ance with the corrections of the text here made. 

84. L 10, t 2 , ma. q j *».« J> for ^yu^ '&***? , g* y ma. q ^ >mJ > ^ Si* 
— A collation of the whole ms. from which our extracts are made 
ia necessary to verify thia statement. Some of the numerals indi- 
cating the numbers of sections are obscurely written in the ma. 
which we have in our handa ; and, though our correspondent's 
analysis haa given us certainty in some of the doubtful cases, it 
still remains uncertain whether the number of sections in chh. I 
and 3 of lect, 4, chh. 4 and 10 of lect 6, and ch. 4 of lect. 7 is 
jr L e. 8, as stated, or p L e. 8. We have also doubted whether 
to read \ L e. 7, or & i. e. 4, for the number of sections in ch. 6 
•of lect 8 ; and what value to assign to a character, repeatedly 
used, which resembles the letter £• In our ms. of the table given 
on pages 78, 74, the same character is used for 0, but of course 
thia ia not its value in the table of contenta. From its similarity 
to the Indian numeral for 4, and because in one instance the letter 
,> seems to be added to explain it, we have assigned to it that 
value. On the grounds assumed, the total number of sections 
cornea out larger, by twenty-one, than the statement of our ms. 

86. 1. 6, h*, ma. ^y&— «ee note a, p. Ill ; L 7, i«, ma. ijj*U— 
aee note 8, p. Ill ; L 9, J* f J^l omitted in ma. 



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Notes on the Book of the Balance of Wisdom. 100 

80. L 6, *», ms. a1>Uo 3 ; 1, 1Q, I*, ma. j*y>. 

88. L 8, m*, m*.jfj>. 

38. L 10, n*, ma. L^oct. 

86. L 2, o 2 , ma. U$»fc?rl3« 

36. 1. 4, f>*, ma. a*j^ ^ t a fragment of a sentence, -which we hare com- 
pleted as in the text : *^> j*&* "i **** /^ .j**** <j»-«L^ 
in accordance with the French translation of our correspondent. 

87. 1.2; ««, ma. |»UU; L 8, r«, .ms. la>feet, . 

88. 1. 6, •», ma. M. |a . ma- *U» ; L % •»«. m. ***». «*, an, j^ ; 
L 8, tr« p ma. K*ttl • *• 9» * 2 » ms- W, U a , ma. ^j L 10, * 2 , ms. 



43. L12, «*, mm. la*^\ 
46. 1. 9, A 9 , ma. LpUm^>, 

48. L 3, *», ms. SC^a^JI ; 1. 4, «P, ms. SJ^li, 

49. L 9, *>, *Ut JJI3 ^H J^**Ut JJ^Jt UT supplied to complete 
the sense. 

51. L 2, r 8 , ms. LfJU; 1. 6, s , ms. XLdttft — by an oversight of the 
copyist, XLd&Jt and *JujJ* in this sentence were transposed ; 1. 7, 
A 8 , ms. Kj^JI ; 1. 12, <», ms. JJ3I ; 1. 16, J», JcJl X^JI jj &l 
^L&pfy t^vj ^5 la^ v^ supplied to complete the sense ; 1. 16, 
**, ms. ^1, 

62. L4 f ^nii.^B u ^y3^tflM 

55. L 16, a* 3 , ms. j£* 3 o!j . 

69. 1. 4, »•, ms. *U , •% ms. {jcj\ , j**, ms. u J^Jt 3 vjly&Ji 3 * , 3 s , 

8L 1.7, r», ms. **?j*. 

98. L 2, •», ms. siyaJskyX ; 1. 7, <», ma. ^^A^XJJ ; 1. 9, A^jjJt* 

supplied to make out the sense. 
64. L 12, at a , ms. lpL*>! . 

66. 1. 3, C, ms. K^aJt ; 1. 12, to 3 , ms. ^L&mXJI ; 1. 17, *', ms. tttt , 
66. 1. 1, y», ^l^II J^AaJt omitted in ms. 

69. 1. 7, *», ms. ^^ksOJt ; L 10, «*, ms. ^ ^; 1. 13, ft*, ms. L$t, 
TO. L 5, c 4 , ma. -WjV> I 8, «**, ms. v'/ > e *> "»• *!>^i L 9, r*, 

ms. jp^P. 



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110 Committee of Publication, 

Page. , 

73. 1. 2, **, ms. Lgjlo> ; 1. 8, i*, ms. L^ ; 1. 6, j* ? ms. ^ ; 1. 8, 
**, ms. ^l&UwJt . 

75. 1. 1, I 4 , before £j\fi UlA\ ms. has *U! {jJlA* £ ^^ f*»*W 
^Pj L&5 [j£>y>\ £La j\iX&A} . But the second division of the book 

commences -with the fifth lecture — see p. 17 ; and . . . (j*Ia&* J, 
is evidently a blundering anticipation of the title of this fourth 
chapter of the third lecture. 

76. 1. 1, «•*, ms. Ju: ; 1. 4, n*, ms. jUiLS ; 1. «, ©*, VV** substi- 
tuted for a word illegible in ms. ; 1. 7, p«, ms. f a1v1 X ; 1. 12, £*, 
ms. (jy*M3^ ; 1. 18, r*, ms. b'jAe <j**J> , •*, ms. «L*. 

77. 1. 1, f *, ms. ot^tt ; 1. 13, t**, ^y^ omitted in ms. ; 1. 14, t>*, for 
^3^ g%^ wft^a.^ -£jW ****$ ms. has ^U^l ^Jixj XajU^ 
Ju3.^ ; 1. 15, tr*, ms. sU , &*, ms. jAK. 

78. L 1 9 y 4 , ms. <jl#U ; **, ms. J*3U* ; a 6 , ms. 1^>U ; L 8, ft 5 , ins. 
Ivirool*; L4,c«,ms. W^i; 1. 6, «t», ms. ^L**! . 

87. 1. 33, «*, ms. ^UAW. 

89. 1.5, r 5 , ms. *iS3. 

90. 1. 6, IT 5 , ms. jJUJt ; 1. 7, **, ms. Bf^t. 

92. 1.0, i 6 , ms. ^c jLoLS — This correction is required by the mul- 
tiplication together of the numbers of the incidents combined. 
The enumeration just made involves nine specifications relative to 
the position of the axis, covering the two cases of separation and 
connection between the tongue and beam, and also the two cases 
supposed with regard to the position of the tongue when joined to 
the beam ; and this number nine is multiplied by the number of 
the specifications respecting the position of the line of suspension 
of the bowls. 

93. J 5 , ms. '*sS ; * 6 , ms. ^y/Juj* ; I 6 , ms. v T ^JLS^, 

97. 1. 15, *•*, ms. £*jLiJt ; 1. 18, n*, ms. UJL*. 

98. 1. 20, ©*, ms. j^tjJt. 

99. 1. 5, jp 5 , Li" omitted in ms. ; 1. 6, 9 s , *+>j conjectural for an 
abbreviation of the ms. 

100. L 3, r*, ms. D ? «AS» ; 1. 7, #*, ms. />\j£ ; I*, ms. ;L*U 3 I . 
103. 1.4,ts»,ms.L# ; 3!, t>6,ms.l?Juul; 1. 6, tt>», for &M £>\ ^$0^ 

ms. has L 5v>jlJI £\ ^/^ 5 1. 12, * 5 . ms. ^Jfr** . 

N. B. Some necessary changes of diacritical points are not here no- 
ticed. The original ms^ it will be remembered, is without these points. 



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Notes on the Book of the Balance of Wisdom. Ill 

II. Notes on Translation and Analysis. 
Referred to by Numerals. 

1, p. 20. The length of the cubit, eJ yXJl , was somewhat variable. 
We read (I.) of iXjJI p|;<3, the hand-dhira', of the Spanish Arabs, meas- 
uring five oUoo , fists, each kabdhah of four f^Lo' ? fingers ; (2.) 
of the dhira* called *^l&Jt — cubitus a situl& cujus magnitudinem 
aequat ita dictus, as Oasiri says — used in Spain, which measured six 
kabdhahs ; (3.) of jd>la!t <A-JI c\jO , the exact-hand dhira', used in 
the East, having the same length as the last named; (4.) of gj/^tt 
^L^wJt , the black dhira', so called because, as is said, its length was 
determined by that of the arm of a slave of 'al-Mamun, measuring six 
fists and three fingers, and by which were sold the byssus and other 
valuable stuffs of the bazaars of Baghdad ; and (5.) of the dhira' called 
jU*&L$3t or Xa&U , of Persian origin, measuring one and a third of 
No. (3.), that is, eight fists. Our author elsewhere speaks definitely of 
cXJI cl ; 3 , by which he probably intends iJj>l*J! v>*J! c! ; 3 , and of 
the dhira' of clothing-bazaars. What he calls the dhira', without 
qualification, is probably to be understood as No. (5.). See Casiri's Bibl. 
Arabico-Hisp., i. 365, ff.; and Ferganensis . . . Elem. Astron. op. J. Golii, 
pp. 73, 74. 

9, p. 24. This term is explained by the figure of the XiU> given on 
page 97. 

3, p. 25. Having satisfied ourselves that M. Khanikoff 's conjecture as 
to the authorship of the work before us is incorrect, we propose simply 
to give the substance of it in this note, in connection with what seems 
to us to be the true view. But we will first bring together a few notices 
of learned men whom our author speaks of as his predecessors in the 
same field of research, who are not particularly referred to in M. Khani- 
koff's note on pages 24, 25. 

Sand Bin 'All is characterized by an Arab author quoted by Casiri, 
in Bibl. Arabico-Hisp., i. 439, 440, as follows : " An excellent astrono- 
mer, conversant with the theory of the motion of the stars, and skilled 
in making instruments for observations and the astrolabe. He entered 
into the service of 'al-Mamun to prepare instruments for observation, 
and to make observations, in the quarter called 'ash-Shamaalyah at 
Baghdad; and he did accordingly, and tested the positions of the 
stars. He did not finish his observations, on account of the death of 



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112 Committee of Publ£cation } 

'aMtfamftn. Willi him originated a well known astronomical table, 
which astronomers male use of to our day. Having been a Jew, he 
became a Muslim by the favor of 'al-Mamun. Several well known works 
on the stars and on arithmetical calculation were written by him." 

Respecting Tuhanna Bin Yusif, Casiri, Id., i. 426, quotes the following 
from an Arab author : " Yuhanna the Christian presbyter, Bin Yusif 
$in 'al-Harith Bin 'al-Batrlk Was a savant distinguished in his time for 
leotoHng on the Book of Euclid, and other books on geometry. He 
made translations from the Greek; and was the author of several 
Works." 

Ibn 'al-Haitham of Basrah, whose full name was 'Abu-' All Muham- 
mad Bin 'al-Hasan Ibn 'al-Haitham of Basrah, as we are told by Wfts* 
tenfeld in his Oesch. d. Arab. Aetate u. Naturfbrscher, pp. 76, 77, was a 
good mathematician as well as skilled in medicine. He rose to emi- 
nence in his paternal city of Basrah, but, on the invitation of the Fatim- 
ite Khallf 'al-Hakim, A. D. 996-1020, went to Egypt to execute some 
engineering, for the irrigation of the country when the Nile should rise 
less high than usual. In this undertaking he failed. The latter part of 
his life was devoted to works of piety and to authorship. He died at 
Cairo, A.H. 430, A*D. 1038. Our ms. gives him the title (jyojl, 
but* as he was generally called from his native city, and the other title 
might so easily be an error of the ms., we have altered it to <,£*>**& . 

From an Arab author, again, quoted by Casiri, Id., i. 442, 443, we 
derive the following notice of 'Abft-Sahl of Knhistan: "Wijan Bin 
Wastam 'Abu-Sahl of Knhistan was a perfect astronomer, accomplished 
in knowledge of geometry and in the science of the starry heavens, of 
the highest eminence in both. He distinguished himself under the 
Buwaihide dynasty, in the days of 'Adhad 'ad-Daulah [A. D. 949-982 — 
see Abulfedae Annates Musi. ed. Reiske, ii. 454, 550]. After Sharf 'ad- 
Daulah had come to Baghdad, on the expulsion of his brother Samsam 
'ad-Daulah from the government of 'Irak [A. D. 986 — see Abulf. Ann, 
ii. 560], he ordered, in the year 378, that observations should be taken 
on the seven stars, in respect to their course and their passage among 
their Zodiacal signs, as 'al-Mamun had done in his day, and he com- 
mitted the accomplishment of this task to 'Abu-Sahl of Knhistan. 
Consequently, the Tatter built a house within the royal residence, at the 
end of the garden, and there made instruments which he had contrived, 
and afterwards took observations which were written out in two declar- 
ations, bearing the signatures of those who had been present, in affirm- 
ation of what they had witnessed and were agreed in." 



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Notes on the Book of the Bolanoeqf Wisdom. l\% 

To this supplementary note we will only add that we e*uld not hfWr- 
tate to translate the name of the person to whom M enelaus is said U* 
have addressed one of his books, which our correspondent feilod ty. 
identify, namely (j^^bl-*}*} , by Doraitian. As the emperor Peinir 
tisn reigned from A. D. 81 to 06, Menelau* most have freea living in 
Ids time. 

Respecting the authorship of the Book of the Balance of Wisdotn, 
after observing that, although the dedication proves it to have been 
Composed at the court of the Saljuke Sultan Sanjar (who reigned over 
a large part of the ancient Khalifate of Baghdad from A.D. 1117 tp 
1 J&7), the recent developments of the history of the Saljukes by DefrG- 
mery afford no clue to the identification of the author, our correspond- 
ent quotes a passage from Khondemlr's Dustur 'al-Wuzara' which he 
thinks may possibly allude to him, as follows : " Nasir 'ad-D!n Mahmud 
Bin Muzaffar of Ehuwaraem was deeply versed both in the science? 
founded in reason and in those based upon tradition, and was especially 
able in jurisprudence after the system of 'ash-Shafff ; at the same time he 
was filmed for his knowledge of finance and the usages and customs, of 
the public treasury. He was the constant protector of scholars and 
distinguished men. The E&^hi 'Umar Bin Sahlan of Sawah dedicated 
to him his work entitled Masa/jr-i-Nasiri, on physical science and logic. 
In the Jawimi' 'at-Taw&rikh it is stated that Nasir 'ad-Dln commenced 
bis career as secretary of the administration of the kitchens and stables 
<rf Stflt&n Sanjar, and that, as he acquitted himself creditably in that 
office, the Sultia named him secretary of the treasury of the whole 
kingdom* and he reached at length the high dignity of Wazir, but, on 
account of the modesty common to men of studious habits, and which 
was native to hifla, he .could not properly perform the duties attached to 
It The Qul^n Accordingly discharged hiin from it, and again entrusted 
to him the administration of the finances, which he transmitted to his 
son Shams 'ad-Dln 'All." On this passage M. Ehanikoff remarks : 
* I do not pretend by the aid of this passage to establish irrevocably 
that Nasir 'ad-Dln is the author of the treatise before us. But his 
feeing a Khuwfcrazmian accords with what our author says of the place 
\Wiere he made his researches ; his participation in the administration 
of the finances would explain his having composed a work for the king's 
treasury ; and lastly, the positive testimony of history as to his erudition 
. . . and the dedication to him of a work treating of physics give some 
probability to the supposition that he may have occupied himself with 
the subject The absence of any direct notice of this treatise on the 
balance in his biography may be ascribed in part to the predilection of 
Khondemlr for politics rather than literary history, in consequence of 
wou vi. 15 



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11% • • • * CkmmUke of Publication, 

which he rarely mentions the scientific labors of those whose memoirs 
he gives, and partly to the circumstance that a work destined for the 
royal treasury, like official reports of the present day, might remain a 
long time unknown to the public/' 

We have thought it proper thus to give the substance of our cor- 
respondent's conjecture. But there can be no doubt that, in the ex- 
tracts from the Book of the Balance of Wisdom which M. Khanikoff 
has given us, the author names himself three times, though in so modest 
a manner as scarcely to attract attention. Instead of heralding himself 
at once, in his first words, after the usual expressions of religious faith, 
as Arab authors are wont to do, he begins his treatise by discoursing on 
the general idea of the balance, with some reference, as it would seem, 
to the Batinian heresy, which gave so much trouble to the Saljuke 
princes, and then simply says.* " Says 'al-Kh&zinl, after speaking of the 
balance in general . . * — see p. 8, and proceeds to enumerate the advan- 
tages of the balance of wisdom, so called, which he is to describe and 
explain in the following work. Farther on, after a section devoted to a 
specification of the different names of the water-balance, and to some 
notices of those who had treated of it before him, he begins the next 
section thus : "Says 'al-Kh&zinl, coming after all the above named . . ." — 
*ee p. 14, and goes on to mention certain varieties in the mechanism of 
the water-balance. The form of expression which he uses in the latter 
of these two passages implies that 'al-Kh&zinl is no other than the 
author himself; for Arabic usage does not allow <^&i to be employed 
to introduce what one writer quotes from another, though nothing is 
more common than for an author to use the preterit JlS , with his name 
appended, to preface his own words. Besides, if 'al-Kh&zinl is not our 
author, but one of those from whom he quotes, who had previously 
treated of the water-balance, why did he not name him in the section 
appropriated to the enumeration of his predecessors in the same field 
of research? In the tide to a table which our correspondent cites in 
the latter part of his analysis, we read again : " Table etc. added by 
.'aloKhaxM" — see p. 60, which also intimates the authorship of the 
work before us, for the writer introduces that table as supplementary to 
one which he cites from another author. Yet farther, if our author's 
name be really 'al-Kh&ainl, his statement respecting the destination of 
his work for the royal treasury — see p. 16 — accords with his own name, 
for 'al-Kh&zinl signifies " related to the treasurer/' and, as M. Khanikoff 
well observes, " the Orientals show as much jealousy in affairs of state 
aa in their domestic concerns." 

Who, then, is our 'al-Kh&zinl f Though unable to answer this ques- 
tion decisively, we will offer some considerations with reference to it 



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Notes on the Book of the Balance of Wisdom. 115 

Possibly our author is the individual of whom I^Herbeiot: makes this 
record : " Khazeni. The name of an author who invented and described 
several mathematical instruments, of which he also indicated the uee"~*- 
aee Bibl. Or., p. 504. Or he may be the same as 'Abu-Jafar 'al-Khaxtn, 
of whom an Arab author, quoted by Casiri, says : "'Abfc-Jafar 'al-KM- 
arin, a native of Persia, distinguished in arithmetic, geometry, and the 
theory of the motion of the stars, conversant with observations and 
their use, famed in his day for this sort of knowledge. He was the 
author of several works, among which is the Book of the Z&j-'as-Safa'j^ 
(^oUjwaJ! .osj: ub^ ), the most eminent and elegant work on the sub- 
ject, and the Book of Numerical Theorems ( XjAXaII JoL«iXf \J*S )" — 
see BibL Arabico-Hisp., i. 408. Yon Hammer, apparently on the au- 
thority of S6dillot, fixes his death in A.D. 1075 — see Literaturgesch. d. 
Araber, vi. 428. Or our author may be identical with Amazon* a person 
long known by name as the author of a treatise on optics translated by 
Bisner, and published at Basel in 1572. It is also possible that one and 
the same individual is referred to under these several names. Eisner 
intimates to us the original title of that treatise on optics in these words : 
u et ut inscriptionem opens, quae authori est de a*pectibti9 y graco, concin- 
niore et breviore nomine opticam nominarem ;" and we hoped to be 
able to obtain from Haji Ehalfah's lexicon some information respecting 
other works by the same author, whioh should throw light upon the 
authorship of the work before us. But this clue to a reference proved 
insufficient, and after several fruitless searches we have not found any 
notice by Haji Khalfah of the famous optician. As to the period when 
Alhazen lived, Risner declares himself ignorant, bat supposes that it 
was about A. D. 1 100 : it will be remembered that our author wrote in 
1121. That our 'al-Khazin! was a native of Persia, as is asserted of 
'Abu-Jafar 'al-Khazin, there is some reason to suppose, from his occa- 
sional use of Persian words ; and here it may be well to observe that it 
is only by an error that Alhazen the optician is made a native of Basrah : 
the error is to confound him with Hasan Bin 'al~Hasan Bin 'al-Haitham 
of Basrah, which has been widely spread, though corrected by Montuda 
and Priestley — see Gartz, De Interpp. et Explanatt Euclidis Arab., p. 22. 
The subject of the work before us is one which the Arabs were accus- 
tomed to class, with optics and other sciences, under the general head 
of geometry — see preface to Haji Khalfah's lexicon, ed. Fiuegel, i. 35 ; 
and there is, indeed, a little sentence in our author's introduction, which, 
with reference to the time when it was written, would seem even to 
betray a writer addicted to philosophizing on light : "For the essence 
of light is its being manifest of itself, and so seen, and that it makes 
other things manifest, and4s thus seen by"— see p. 7* Again, the style 



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116 Cbmmtoe* of Publication, 

fcf 'Abu-Jato 'al-Khirtin, as indicated ifi an extract firom <me of th* 1 
of the Bodleian Library, given in Catal BibL BodL, ii. Wll, aft follows : 

£^ ^> i*U ^ ^ CT JtXtfil JJJjjj i*W ^LiAdt ^ *ui yaXfy 
JbC&^t 1 " he aimed at brevity in the work, abridging the phraseology 
and diminishing the number of the figures, without removing doubt or 
doing away with obscurity," which refers io a commentary on Euclid, 
seems to us very like that of our author. 

TJpon the whole, we incline to believe that our author and 'Abu-Jafar 
'al-Khazin and the optician Alhazen, perhaps also D'Herbelot's Khazeni, 
are one and the same person. We venture, at least, to suggest this, for 
confirmation or refutation by farther research. We may here Bay that 
We have been in some doubt whether to read the name of our author 
'al-KhAzin or 'al-Khazini, the Arabic ms. sent to us by M. Khanikoff, 
Which gives the latter reading, not being decisive authority on this 
J>oint We beg our correspondent to settle the question by reference to 
the original ms. For the proposed identification, however, it is equally 
Well to assume the name to be 'al-Khazint, inasmuch as, according to 
Bisner, the author of the book on optics was called "Alhazen fiL 
Albayzen," that is, 'al-Khazin Bin r al-Khazin, to which 'al-Khaxinl, in 
the sense of " related to 'al-Khazin," is equivalent 

4, p. 87. In reproducing this figure, we hare struck out ft point 
assumed in the original between h and £, and made use of in the fiftk 
theorem, because it is of no service, and only makes the figure inoefc- 
eistent, and less applicable to the following theorem*, 

6> p. 39. It seems to us that the remark of our author here referred 
to is misinterpreted by M. Khanikoff* The former means simply to aay 
that no interference with one another's motion is apparent in the case 
of the celestial spheres, while he neither affirms nor denies the principle 
of universal gravitation* 

6, p. 42. The figure here given of the areometer of Pappus is con- 
siderably altered from that presented by M. Khanikoff. The latter 
occurs twice in the manuscript, once in the Arabic text, and once in the 
translation : the two forms are of somewhat different dimensions, and 
are quite inconsistent with one another with respect to the details of the 
graduation, which are moreover, in both, altogether inaccurate. Both, 
however, agree in offering, instead of a double scale, two separate scales, 
standing in a reversed position at a slight interval from one another. 
As it was impossible to reconcile such a figure with the directions given 
>n the text^ we have preferred to construct a new one, in accordance 
with our best understanding of those directions. In so doing, we hav* 



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Notes on ike Book of (he Balance of Wisdom. 



117 



npeBed to aaesbd the text slightly as regards tile letfcsrtd- points 
referred to, a* Weil a» Ike lettering of tike figure itself; inasmuch as the 

latter in the manuscript 













orft 








€ 




O 






Qfc 




cf 








&: 










at 




$ 


i 







so imperfect and inaccurate 
as to be unintelligible. For 
the purpose of showing the 
alteration we have made, we 
present herewith an exact 
copy of the portion of the 
manuscript figure about the 
equator of equilibrium. It 
will illustrate also the manner 
in which the numbering of the 
instrument is indicated (very inaccurately upon the left scale) in the figure, 

7 9 p. 42. Bee the statement, on page 98, that the Arab physicists 
were accustomed to enchase silver points upon the right arm of the 
beam of the balance! Ibr specific gravities ; and the description of the 
balance on page 97. To graduate by round points seems to have been 
Hie mode among the Arabs. 

•, p. 47. We have constructed this table anew, and have corrected 
at many points the sixtieths of our original : in some cases, the readings 
of the latter may have become corrupt ; in most, there was probably a 
want of accuracy in the original constructor. 

The readings of the manuscript which we have altered are as follows: 



tfNum** 


«, Ftu. 


£istieffcf. 1 


LimqfNumton. 


PmU, 


JSvctktl 


IDS 


96 


16 


87 


114 


68 


104 


96 


12 


86 


116 


12 


103 


97 


9 


83 


120 


30 


102 


98 


6 


76 


131 


36 


101 


99 


3 


69 


144 


54 


99 


101 


2 


64 


166 


1* 


98 


102 


6 


62 


161 


20 


97 


10* 


S 


61 


163 


58 


96 


104 




$6 


181 


60 


94 


106 


21 


52 


192 


20 


88 


113 


37 









Here are also several esses in which we have retained the figures 
given by the manuscript, although not quite correct, when they differed 
by less than 1 from the true value. Thus, opposite the number 106, we 
find set down a remainder of 21 sixtieths, while the true remainder is 
20.876, and, of course, 20 would have been more accurate. 



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118 Gmmtoee <tf Publication, < 

0, p. 50. We alone are responsible for the translation of tbi» settlon, 
our correspondent having left it untranslated. In the lettering of the 
first figure here introduced, we have so for deviated from our Arabic 
original as to substitute g and p for £ and <J«, respectively, in order 
to simplify the transcription. 

10, p. 53. The meaning of this somewhat obscure statement is prob- 
ably as follows. Two heavy bodies at opposite extremities of a lever 
act upon each other by their gravity to. produce motion, and remain at 
rest only when their common centre of gravity is supported. The same 
is true of balls thrown into a spherical vase : they act upon each other 
by their gravity to produce motion, and they remain at rest only when 
their common centre of gravity is supported, that is, when it stands 
over the lowest point of the spherical surface. 

11, p. 54. In our reproduction of this figure, we have reduced its 
size one half, improved the form of the " bowl," /, and given in the 
margin the explanations which in the original manuscript are written 
upon the figure itself at the points where the letters of reference are 
placed. 

13, p. 56. We have taken the liberty of slightly altering our cor- 
respondent's manuscript in order to insert the table here given, because 
the latter seemed to be so distinctly referred to upon page 78 that it 
was necessary to assume that the Arabic work originally contained it 
M. Khanikoff gives the mean weight for bronze as 11 m., £} d. ; but, 
considering the acknowledged corrupt state of the manuscript in this 
part, we have thought ourselves justified in amending the reading to 
11m., 2 d., since this value is required by that of the " result n derived 
from it for the succeeding table. In the table on p. 78, it will be no- 
ticed, bronze is omitted. 

13, p. 63. 'Ahmad 'at-Taifeshl, in his book on precious* stones, ed. 
Raineri, page 10 of text, describes the y&kut 'asmfcnjunl as including 
"the cerulean, that which resembles lapis lazuli, the indigo-like, the 
collyrium-like, and the dusky" — by which may be intended, as the 
editor says (annott, p. 80), all sorts of sapphire and aqua marina. 

14, p. 63. According to 'at-Taifashl, page 13 of text, the raihanl, or 
basil-like, is a variety of the emerald, " of pale color, like the leaves of 
the basil ;" the same authority defines the silk!, or beet-like, as another 
variety of this mineral, "in color like the beet," that is, probably, like 
beet-leaves. 



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Notes on the Booh of the Balance of Wisdom. 119 

16, p. 64. In thus rendering the word \j*\jk& , we conjecture it to 

be derived from H suft with the signification " circulus ungulas bubulse 
magnitudine, w as given by Freytag. Raineri gives us another reading 
for this word, namely ^tiuJi, and explains it as possibly signifying 
of Bukhdrd — see text of 'at-Taif&shl, p. 35, and annott., p. 100 — 
which, however, seems to be quite impossible. The Arab mineralogist's 
description of the species of onyx bearing this name is as follows : 
"i^juit is a stone composed of three layers: a red layer, not trans- 
parent, followed by a white layer, also not transparent, next to which is 
a transparent, crystal-like layer. The best specimens are those of which 
the layers are even, whether thick or thin, which are free from rough- 
ness, and in which the contrast [of color] and its markings are plainly 
seen ;" which corresponds to what Caesius, in his Mineralogia, Lugd., 
1636, p. 569, says of the most precious sardonyx, thus : " Quaeres tertio 
quaenam sit sardonyx omnium perfectissima. Respondeo esse illam quae 
ita referat unguem humanum carni impositum ut simul habeat tres col- 
ores, inferiorem nigrum, medium candidum, supremum rubentem. . . . 
Nota autem, cum sardonyx est perfecta, hoB tres colores esse debere 
impermixtos, id est, ut zona alba nihil habeat mixtum alieni coloris, et sic 
de nigrfc et purpurea." But, if our reading ^'yM' is correct, and the 
derivation which we have suggested for the word is adopted, this species 
of onyx must have been so named from specimens with their layers of 
different coIots intermingled, which the modern mineralogist would call 
by the name of agate rather than that of onyx. Perhaps, however, 

the reading should be ^ JyuJI , from ^jSaJ! , the name of a certain 
plant, which we have not identified. In this case, the name would be 
similar in its origin to " basil-like" and " beet-like," applied to varieties 
of the emerald, and appropriate for specimens of onyx with either dis- 
tinct or intermingled layers, the veining of the mineral having nothing 
to do with its name. 

16, p. 65. Caesius, Id., p. 520, says : "Dioscoridea, Judaicus, inquit, 
lapis in JudasA nascitur, figur& glandis, eleganter et concinne confectus, 
lineis inter se aeqnalibus veluti torno factis," etc. Prof. J. D. Dana of 
Yale College, to whom we are indebted for the foregoing quotation, 
infers from this description that the Jews 9 stone was the olive-shaped 
head of the fossil encrinite. Ancient physicians dissolved it for a 
draught to. cure gravel 'Ibn-Bait&r, in his Mufrid&t, ed. Sontheimer, 
i 285, speaks of it in the same terms as Caesius does. 

1*, p. 65. Probably a fossil. M. Khanikoff quotes the following from 
the Kfcnros : £)\>*3$ *** i>Sj*$ Wj . . , gaJJl '**££ K^i Xjb glfcyJJ 



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120 Committee of Publication, 

^£^U, M The crab is a river-animal of great dm ... on the other kind, 
the sea-crab is a petrified animal." One might ask the question, whether 
the latter was so called because found in the sea, or whether that name 
implies a belief that the sea once extended where afterwards was dry 
land, in accordance with modern geological discovery, 

1§, p. 66. In explanation of this term, onr correspondent cites, from 
a Jaghatai Turkish translation of 'al-Kazwtnl's 'Aja'ib 'al-Makhlukfrt, a 
passage which states that "the finest quality of glass was Pharaoh's 
glass, found in Egypt" The original Arabic gives no such statement 
under the head of glass. For an account of the translation referred to, 
see M. Khanikoff in the Bulletin Hist-Phil. of the Imperial Academy of 
St Petersburg, for Nov. 8, 1854 (Melanges Asiatiques tirts du BoUetin 
eta, ii. 440-446). 

19, p. 72. We have # substituted this citation from the original Arabic 
of 'al-Kazwinl for a passage which our correspondent here quotes from 
the Jaghatai Turkish translation referred to above* The citation of M. 
Khanikoff of which only the first sentence is recognizable in the Arabic, 
is as follows : " Khuw&razm is a vast, extensive, and populous province. 
There is in it a city named Jurj&niyah, . . . The cold there is so intense 
that a man's face freezes upon bis pillow ; the trees split by reason of 
the cold ; the ground cracks ; and no one is able to go on horseback. 
One of its frontiers is Khurasan, the other Mawaralnahr. The river 
Amu [the Jai^un], freezes there, and the iee extends from there to the 
little sea. In the spring, die waters of the little sea mingle with the 
water of the Amu, and come to Khuwarazm f on which our correspond- 
ent observes : " If I am not mistaken, this passage, which establishes 
with certainty that the waters of the Amu reached the Sea of Aral only 
during the spring freshets, is unique ; and it points out to us, perhaps, 
the way in which the change of its mouth originally took place. n 

By way of illustration of 'al-Kazwlni's description of the lower course 
of the Oxus, it may not be amiss to cite what Burnes says of it in his 
Travels into Bokhara, iii. 162. Having spoken of its winding among 
mountains till it reaches the vicinity of Balkh, Burnes says : " It here 
enters upon the desert by a course nearly N. W., fertilizes a limited 
tract of about a mile on either side, till it reaches the territories of 
Orgunje or Khiva, the ancient Eharasm, where it is more widely spread 
by art, and is then lost in the sea of AraL In the latter part of its course, 
so great is the body of water drawn for the purposes of irrigation, and 
so numerous are the divisions of its branches, that it forms a swampy 
delta, overgrown with reeds and aquatic plants, impervious to the hus- 
bandman, and incapable of being rendered useful to man, from ifc un- 
varying humidity." 



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Notes on the Book of the Balance of Wisdom. 121 

90, p. 78.* These terms are explained by a passage appended to this 
section in the translation of our correspondent, for which he has omitted 
to give the Arabic text. We quote the translation here : 

" All the substances mentioned in this section sink in water if the 
weight of their equivalent volume of water is less than 2400 tassujs, 
and float if that weight equals or exceeds the same number. 

End of the first section" 

91, p. 74. It might be suspected that the word -^Uft, "ivory," 
should rather be viUxJi , " resin," this being one of the substances of 
which, in the opening of the chapter, our author proposed to give the 
water-equivalent As, however, the specific gravity derived from the 
equivalent given is 1.64, and that of the common resins is only about 
1. to 1.1, it is perhaps easier to assume that resin has been accidentally 
omitted from the table. 



99, p. 76. This process can be made clearer by an algeoraic method 

of statement. Let (7— a cubit, c— the length of a side of the cube 

measured, and (— the diameter of the silver thread : then (7— 4c-f- 

ifc-fO : but <:— 259 < ; therefore, (7— 1082^*- For the fraction ^ 

our author now substitutes ^, because, as appears from their use in the 

table on p. 46, sixtieths are to him what decimals are to us, and -fa are 

64 923 
nearly equivalent to fo C, then, equals — ~ — . Substituting, in the 

proportion c* : C* : : 9415 tassujs : the weight in tassujs of a cubic 

cubit of water, the numerical value of the first two terms, we have 

,*«*«**,> 278,650,180,698,467 ^. %tf flfl/l ^ f0 T . Al _ .. . 

17,373,979 : — J„JL : ■ 9*15 s 686,535.58. If the original 

' ' 216,000 * e 

fraction ^ had been retained, the result would have been 686,525 

tassujs. 

The question naturally arises, why 'Abu-r~Raih&n had recourse to the 

use of the silver thread in making this experiment. It is evident that, 

when once we have the value of C expressed in terms of c, we may 

reject t altogether, and obtain the same result as before by the much less 

, ^ • ^ . *- . , 6,644,672/. , , ,188 *\ 

laborious reduction of the proportion 1 : • - li.e. I 3 : \j~rr) I 

: : 9416 : 686,525. This fact, however, our author does not seem to 
have noticed. Was the silver thread, then, employed merely as a me- 
chanical device for facilitating an exact comparison of C and cf This 
seems by no means impossible, since, after finding the first remainder, 
and perceiving it to be equal to 46 diameters of the thread, the relation 
of that remainder to the side of the cube would be at once determined, 
without going through with the two additional, and more delicate, 
vol. vi. 16 • 



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122 Committee of Publication, 

measurements otherwise required, and exposing the process to the 
chances of error to which they would give rise. We may also, per- 
haps, suppose that the original experimenter was actuated by a desire 
to furnish to any who might repeat the experiment after him the means 
of comparing their results with his, and that, in the want of any exact 
standard of measurement (since the measures may have varied in differ- 
ent localities not less than the weights are shown to have done by the 
table upon p. 81), he devised this method of the fine silver thread with 
the view of providing, in this case, such a standard. 

23, p. 78. We have introduced into this table a number of emenda- 
tions, which were imperatively called for. All the data for constructing 
it had been given before, in the table pf water-equivalents upon p. 56, 
and the determination of the weight of the cubit cube of water just made, 
and we had only to work out, in the case of each metal, the propor- 
tion : the wajer equivalent of the metal, in tassujs : 2400 : : 686,535.53 : 
the weight of a cubic cubit of the metal. For mercury, the manuscript 
gives erroneously 3 8*7,973 m. ; for silver, 294,607 m. ; this would be 
the correct result if the calculator, by a slip of the pencil, had taken 
686,435.53 as the third term of his proportion. For tin, the Arabic 
gives 209,300 ; but, as the translation presents the correct number, the 
former must be an error of M. Khanikoff's copyist The column of 
'istars is. left unfilled in the Arabic manuscript ; our correspondent had 
supplied the deficiency, but incorrectly in the majority of cases (all ex- 
cepting mercury, silver, and iron). It admitted, indeed, of some ques- 
tion how many 'istars our author reckoned to the mann : we adopted 
40, both because that seems to be the more usual valuation, and because, 
by assuming it, the column of fractions of 'istars comes out in most in- 
stances quite correct : only in the case of lead, the manuscript gives a 
remainder of £ + \ ; in that of iron, £ ; of tin, -J- + ■£. The same valu- 
ation of the 'istar was assumed in correcting the reading at the bottom 
of p. 77, where the great corruption of the whole passage, and the 
absurdity of its unamended readings, was very evident 

94, p. 82. Considering the uncertain character of even the main 
elements entering into this calculation, and that its result cannot accord- 
ingly be otherwise than approximate only, it seems to us that our cor- 
respondent might have spared himself the labor of calculating the 
effects of an assumed difference of temperature, pressure, and gravity : 
the modifications which are thus introduced lie far within the limits of 
probable error from other sources ; and, in fact, had these modifying 
circumstances been left out of the account, a result would have been 
.arrived .at nearer to the value which M. Khanijcoff finally adopts for the 



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Notes on the Book of the Balance of Wisdom. 128 

cubit than that which is actually obtained by his process. The value 

c=500 mm., namely, supposes — — .1250; the actual value of the 

latter quotient, after modification of m 3 , is .1291 ; before modification, 
.1287. We have not, therefore, in this instance, been at the pains of 
verifying either the formulas or the calculations of our correspondent 

25, p. 86. In reproducing the figure here given by our correspond- 
ent, we have reduced its dimensions one-third, without other alteration. 

20, p. 86. This figure also is reduced one-third from that given, in 
M. Khanikoflfs manuscript 

27, p. 87. We have here followed our correspondent in giving only 
the original term, not being sure enough of its precise signification to 
venture to translate it, though we think it might be rendered " table of 
plane projections," or "planisphere." The connection shows that it 
denotes some astronomical instrument ; and as well for this reason as 
because \j&f-, and not i^Jus, means "latitude," besides that ^\ is 
not a plural, Casiri is wrong in translating ^i&fiit <s*a\ v^» m a 
passage quoted from him on p. lid, as he does, by "Liber Tabularum 
Latitudinum." 

9§, p. 88. This figure is an exact copy of that given by M. Khani- 
koflf, except that it is reduced to one-third its original size. Its insuffi- 
ciency to explain the construction and adjustment of the parts which it 
represents is palpable. For farther explanation see note 33. 

99, p. 00. Our two diagrams on pp. 89, 91, though faithfully repre- 
senting their originals, are, for convenience, made to differ from them 
in dimensions. 

30, p. 90, Literally "the point going downward," implying the 
conception of a concentration of the weight of a body in its centre of 
gravity. 

31, p. 94. We suppose the centre of gravity spoken of under the head 
of "Determination relative to the Beam connected with the Tongue," 
in this table, to be the common centre of gravity of the tongue and 
beam united. This seems to us to be indicated both by the reading of 
the Arabic text of the table, as it came to us, and by the suppositions, 
respecting the connection of the tongue and beam, considered in the 
extract which precedes it. But some changes of reading were required 
to make the table correct The following fragment shows what are the 



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124 



Committee of Publication, 



specifications of the text which we received, in those places where we 
have made changes. 

Determination relative to the Beam connected with the Tongue, 

[the Axis being] 

at the C. of Grav. above the 0. | .... 



[the Axis being] 
Delow the 0. 



Revers. 



Revere. 



Nee. Parall | Equip. 
Equip. 



Equip. 



Revere. 



Nee. Parall.l Equip, 



On the supposition that the centre spoken of is the centre of gravity of 
the beam, many more alterations would have been necessary. 

39, p. 95. Our correspondent did not translate this passage, so that 
we alone are responsible for the description of the winged bowl. We 
were about to offer some conjectures which seem naturally to suggest 
themselves in explanation of the peculiarities of this bowl ; but, since 
we have so little ground for certainty in regard to it, we prefer to waive 
the subject. It will be noticed that our two drawings of the bowl are 
dissimilar, and that the one which is given in the figure of the whole 
balance is the least conformed to the description of our author. 

83, p. 96. The figure of the Balance of Wisdom given here is as 
exact a copy as possible of the figure in M. Khanikoff's manuscript, 
excepting that it is reduced in dimensions one-third. Where the beam 
is crossed by the two slanting pieces, the shading of the one and the 
graduation of the other are both continued without a break, so that 
it is impossible to decide by the figure which is regarded as lying 
upon the other. This ambiguity the engraver has reproduced as 
well as he could. It is impossible, with no more light than we have 
on the subject at present, to determine the use and connection of all 
the parts of the balance here represented. There is even an appar- 
ent inconsistency between the figure on p. 88 and the drawing of 
the same part here given. The two drawings are, indeed, quite in- 
sufficient of themselves to explain the construction of the instrument* 
But what is especially to be regretted is that M. Khanikoff has 
omitted to cite our author's description of the axis and its pivots, or 
to give us so much as a hint of the mode in which the beam was 
held up. Yet, as the case now stands, a definition in the Kamus of 
the use of the two sloping pieces represented in our drawings on 
opposite sides of the tongue of the balance, enables us to make out 



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Notes on the Booh of the Balance of Wisdom. 125 

some of the leading parts of the construction. Those sloping pieces 
are called by the name of Q^LaJt , substituted for the ms. reading, 
^LaJI, which makes no sense. Of this term the Sihah says: 
?jaJ1 d LJ d UA£j UU1 t ; La%, "the ^Lo are those two 
pieces which show the tongue of the balance." But the Kamus is more 
definite, and says : ^\j^ q^ qU^X<j qIjkXkXs* j*&\j q^LaS^ 
ij&jtf j#Jd *% ^U» *I vi^JUxi *j/j, "q^LaJI with kasr signifies 
two pieces of iron which enclose the tongue of the balance, and Kif 
signifies that I have made for the tongue such two pieces, and that it is 
y^h , which is equivalent to saying that it has stops put to it" Plainly, 
then, the tongue of the balance moved from one to the other of the 
two pieces of iron thus described, and must have been in the same plane 
with them ; and, since the middle vertical line of the tongue would of 
course range with that of its frame, the axis must have been bisected, 
longitudinally, by the same line. Thus, then, we are able to conjecture 
what the two parallel pieces in our larger drawing are intended to rep- 
resent They may be the supports of the two ends of the axis ; and 
the inconsistency between this drawing and the one on p. 88, which we 
have alluded to, may be only apparent. For above the supports of the 
axis, and between the lower ends of the q^Lo, there must have been 
an opening to allow the tongue to play within its prescribed field of 
motion — whether the frame of the tongue rested upon one of the sup- 
ports of the axis, as our large drawing seems to show, or stood between 
the two. In accordance with this view, we regard the figure on p. 88 
as representing only the tongue and its frame, and the bends at the 
bottom of it as bends of the qIjI*& . 

Both of the parallel pieces are represented as if indefinitely prolonged 
towards the right, and, though this may be a mere inaccuracy of draw- 
ing, it is not unlikely that they were attached, in some way, to a fixed 
upright support, on that side, for the sake of giving a more stable posi- 
tion to the axis. 

It remains for us to state the grounds on which we have assigned to 
tiie word Ma^ytSt the signification of " front-piece." This term is in- 
scribed, in both of our drawings, on the top-piece of the frame of the 
tongue, and also, in the larger one, along the nearer of the two parallel 
pieces, at e : so that one might think that two different parts of the struc- 
ture were called by the same name, on account of something in common 
between then ; or else that it was applied to the whole structure consist- 
ing of the frame of the tongue and the two parallels. But we believe 
the inscription of this word along one of the parallels to be simply a 
misplacement, **ty*tt being there the first word of a sentence of ex* 



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126 Committee of Publication, 

planation which is finished under the beam on the left. We therefore 
regard this term as appropriated to the frame of the tongae. The 
etymology and form of the word itself give it the meaning of " that 
which fronts ;" but, unfortunately, neither the Sihah, nor the Kamus, 
nor any European dictionary of the Arabic which we have been able to 
consult, defines it in its application to the balance. 

In making these suggestions we have been aided by a scientific friend, 
Rev. C. S. Lyman of New Haven, to whom we desire to express our 
obligations. 

84, p. 97. So called, we may suppose, from their being obtained by 
calculation. 

35, p. 98. To explain this formula, 
Let W be the weight in mithkals of a compound body (gold and silver) ; 
x the weight in mithkals of the silver contained in it ; 
v>-z " «' the gold " " 

Then *. yr. (the spec. grav. of the compound) — W (its weight in air) 
divided by the weight of water which it displaces. But if d' and d" 
are the specific gravities of gold and silver, the water displaced by 
( W-x) mithkals of gold will weigh ( W-z) -r-d' ; that displaced by * 
mithkals of silver, x-$-d". Hence 

W _ W _f W-x , x\_ W *._* 

W-z. x — *^ ; OT w—\d r ~ + d r ')-- T d'^~d" 9 



By transposition, Zp~~W> :=i ~d r ' mm * whence * = W 

3d, p. 105. See note 5. 



1 


1 


d'~ 


s.gr. 


1 


1 


d r 


~d" 



We have just received, in the Journal Asiatique, v e Serie, xi., 1858, 
a paper by M. Clement-Mullet, which exhibits tables of water-equiva- 
lents, water-weights, weights of equal volumes of substances, and spe- 
cific gravities, derived from 'Abu-r-Raihan, through the medium of the 
Ayin-'Akbari. The foregoing article will be found to correct and sup- 
plement the statements of M. Clement-Mullet's paper, in many particu- 
lars, as, indeed, it rests upon a much wider basis ; and we feel sure of 
its meeting with a cordial welcome from the French savant, whose 
interest in the subject has been manifested by valuable contributions to 



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Notes on the Book of the Balance of Wisdom. 127 

war knowledge of the sciences of the East It may not be amiss, then, 
to indicate here some errors in the tabular statements of the paper 
referred to. . 

Since the sum of the water-equivalent and the corresponding water' 
weight must in every instance be 100 mith^als, the water-equivalents 
given for mercury, silver, and emerald require the water-weights of these 
substances, respectively, to be 92.3.3, 90.1.3, and 63.4, instead of 92.0.3, 
90.1 , and 68.4, as stated. 

The water-equivalent of copper is stated to be 11.3, whereas our 
author gives us 11.3.1 : but the weight assigned to the equal volume of 
copper also differs from our author's statement, being 45.4 instead of 
45.3. This brings us to a point of greater apparent disagreement be- 
tween 'Abu-r-Raihan and 'al-Khazinl, the columns of weights of equal 
volumes. Now, as these weights must in every case be derived from the 
data furnished by the preceding columns, by a rule which our author 
gives, although it is nowhere presented in the quotation from the Ayln- 
'Akbari, we must be allowed to mention the following as errors under 
this head, viz: mercury 71.1.3 for 71.1.1, silver 53.5.1 for 54.0.2, bronze 
46.1.2 for 46.2, copper 45.4 for 45.3, brass 44.5.1 for 45, tin 38.0.3 for 
38.2.2 ; ruby (or red hyacinth) 97.1.1 for 97.0.3, ruby balai (if the same 
as ruby of Badakhshan) 90.2.2 for 90.2.3, emerald 69.2.3 for 69.3, cor- 
nelian 64.3.3 for 64.4.2. 

But, if we examine the statement of the weights of equal volumes in 
Gladwin's translation of the Ayln-'Akbari, we shall find that this may be 
so interpreted as to come into exact coincidence with our author's state- 
ment, in all but one of the cases here in question. For, supposing that 
Gladwin's figure 8, wherever it occurs in the fractional columns, cornea 
from a wrong reading of £ for £,=0, agreeably to a suggestion of M, 
Clement-Mullet, or is a mistake for 3 = ^ ; and farther, that Gladwin's 
Vfigure 5, given for tassujs in the weight of the equal volume of brass, is 
a mis-reading of o for ♦ ; and lastly, supposing Gladwin's mithkal-figure 
4 in 94.0.3 for amethyst (or red hyacinth) to be a mis-reading of J> 
f or j — suppositions which derive support from an extended comparison 
of the tables given by Gladwin with the corresponding tables of the 
foregoing article — we obtain the following weights of equal volumes 
from the English translation of the Ayln-'Akbari, viz : mercury 71.1.1, 
silver 54.0.3 (instead of the true number 54.0.2), bronze 46.2, copper 
45.3, brass 45 ; amethyst (or red hyacinth) 97.0.3, ruby (probably ruby 
of Badakhshan) 90.2.3, emerald 69.3, cornelian 64.4.2. 

We have thus far passed over pearl and lapis lazuli, because the weights 
winch the French savant gives to equal volumes of these substances 
show a double error — their respective numbers being transposed, while, 



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128 Committee of Publication, Notes, etc. 

at the same time, the number truly belonging to pearl is 65.3.2, and not 
65.4. This transposition, though not the wrong number for pearl, 
occurs also in Gladwin's translation ; and in correspondence with it, in 
both the French and the English presentation of 'Abu-r-Raihan's results, 
the water-equivalents of pearl and lapis lazuli are transposed, that of 
pearl being 37.1 for 88.3, and that of lapis lazuli 38.3 for 37.1. I£ how- 
ever, the specific gravity of pearl was supposed by the Arabs to be less 
than that of lapis lazuli, agreeably to our table, its water-equivalent 
must have been rated the highest. It is scarcely to be believed that 
any accidental difference of quality in the pearls of 'Abu-r-Raihan from 
those experimented upon by 'al-Khazinl, could have led the former to a 
water-equivalent for pearl precisely the same as that which the latter 
found for lapis lazuli. 

Again, M. Clement-Mullet's list of specific gravities deviates in several 
instances from the results which should have been brought out by the 
water-equivalents given, as : silver 10.35 for 10.30, copper 8.70 for 8.60 
(according to 'al-Khazini, 8.66), etc. The specific gravity obtained 
for amber, 2.53, while agreeing with the water-equivalent assigned to it, 
39.8, so far exceeds the modern valuation as to occasion a remark by 
the French savant : but our author gives it a much higher water-equiva- 
lent, namely 118, and consequently a much lower specific gravity, 
namely .85. It would seem, then, altogether likely that not amber, but 
some other substance, was here referred to by 'Abu-r-Raihan ; perhaps 
coral, which is given by 'al-Khazinl in the same connection. 

These remarks cover all the substances mentioned by 'Abu-r-Raihan, 
excepting gold, lead, iron, celestial hyacinth, and crystal — in regard to 
all of which there is no disagreement between him and 'al-Khazinl — so 
that, besides their purpose in the way of criticism, they serve to show 
an almost entire identity between 'Abu-r-Raihan's tables, so far as they 
go, and those of our author. The course pursued by the latter, in his ' 
tabular statements, would seem to have been to adopt, with some cor- 
rection, the results obtained by the earlier philosopher — to whom, it 
will be remembered, he frequently refers as an authority — and to add to 
them by experiments of his own. 



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ARTICLE II. 



OBSERVATIONS 

or in 

PREPOSITIONS, CONJUNCTIONS, AND OTHER PARTICLES 

or THS 

ISIZULU AND ITS COGNATE LANGUAGES. 

By Bit. LEWIS GROUT, 

MttBIOWABY OK TBI A. B. 0. F. M. IN BOOTH AFM0A. 



Presented to the Society Nov. 4, 1668. 



1. Na is both' a conjunction and a preposition, signifying and, 
even, also, with; it is also used as an interrogative particle. 
The same word na = and, with, etc., prevails among all the 
cognates of the Isizulu, along the eastern coast of Africa — at 
Delagoa Bay, Inhambane, Sofala, Tete and Sena, Quilimane, 
Mosambique, Cape Delgado; also in the Suaheli, the Nika, 
Kamba, rokomo, and Hiau dialects. It is also common in some 
of the interior and western dialects ; sometimes, however, with 
some modification of form and import ; thus, in the Mpongwe, 
no, ni, n J = with, for ; Benga, na «= with ; Setshuana, na = with, 
and ; as nabo } with them ; naiu } with you. 

Corresponding to the use of na as an interrogative particle, 
which always follows the interrogative phrase or sentence, and 
takes an accent with the falling slide of the voice, the Mandingo 
has di in many cases ; and the Bornu language has ba, originally 
ra / and this ra is the same word which is used in that language 
as a conjunction = or. As the asking of a question implies the 
return of an answer — an additional remark — the indicating of a 
question by the use of a conjunction is not unphilosophical ; nor 
does it differ in principle from the English, which makes most 
questions to end in an elevated or rising tone, thus indicating a 
state of suspense and the expectation of an answer. 

VOL. VI. 17 



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480 X. Grout, 

2. Nga is a preposition, signifying by, through, by means of, 
oir account of, in respect to, at, with, toward, near, about. The 
Setshuana has $w»at> with, concerning, of, from, respecting; 
and the Mpongwe, <70 = at, upon, to; gun ~ at, in, from; and 
gore = for, to, at. Some of the uses of ga in the Setsh uana would 
seem to indicate its cottespoftdence to the Zulu ka; and some of 
the uses of go and gwi in the Mpongwe are closely related to the. 
use of ku in the Isizulu. 

8. Ku, the preposition =* to, from, in, with, is not only used as 
a separate word, but enters as a prefix into the composition of 
several adverbs, especially those which are formed from adjec- 
tives* With the initial u, it forms the sign of the infinitive ; thus, 
uku tanda, to love. Ku is also found in many of the cognates 
of the Isizulu, both as a mark of the infinitive, and as an ele- 
ment in the formation of advefbs. Thus, in the Suaheli, ku 
nena, to speak ; kufania, to make ; Maravi, ku lira, to weep ; 
Cape Delgado, Tete and Sena, ku rim; Inhambane, ku lila — to 
weep ; Mpongwe, go kamba, to speak ; Setshuana, go bofa, to 
bind. So in adverbs, Tete and Sena have kunsha, Cape Delga- 
do and Sofala, kwndsha, while the Inhambane has papandsJie, all 
in the sense of the Zulu panhle, without, outside. So again, Tete 
and Sena and Quilimane have kuzogoro and padzugoro = before; 
Tete and Sena, kwmbati and pambari^ by the side; kuzuru t 
above ; kumhuio, after ; kuno, on this side, 

4. Ka. The particle ka, as seen in many adverbs, is originally 
a preposition — the genitive a hardened by k; thus, kakulu (of 
siae), greatly ; kaloku (of this), now ; kanye (of one), once. The 
same is found in the Setshuana as a preposition separable = for, 
by, in, with ; and sometimes inseparable, and in the softer form 
aa; thus, gangue, once; gaberi, twice ; gantsi, frequently; and 
in some words, as haguluu, the sound of g is reduced to"a mere 
aspirate. The Setshuana has the preposition ka as separable, 
in some cases where the Isizulu has the inseparable pa; thus, 
Setshuana ka pek =» Isizulu pambili, before ; ka intk — panhfy 
without; kagare eepakati, within. We find ka used in the Tete 
and Sena as in the Isizulu; thus, habozi, kaposi, once; kavire-konz* 
or kabiri-konzi, twice; kaviri-kaviri, always. So in the Inham- 
bane, karinif how? karora f thus* 

5. Kwa is evidently composed of ftw=to, from, + the genitive 
a, the sign of source, possession, and designation ; hence the gen* 
eral signification, to, from ? of, at, with, in — its more specific im* 

r>rt being determined by its connection ; thus, ngi ya kwa Zulu, 
go to the Zulu country ; ngi vda kwa Zulu, I come from the 
Zulu country; abantu ba kwa 'Must Musi's people, or the people 
are with Musi, are at his plaoe, or they belong to him, according 
to the connection in which the phrase is used. So in the Suaheli, 
nime kuenda km Wa&, I Wfcfat to the Governor; Mpongwe, 
agendaga gwi ^longa, he went to (the) country. 



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Particles ofihclstzulu, etc. 131 

6. Pa. The inseparable particle or preposition pa=*cl<fee, 
near, by, at, in, among, which enters into so many of the ad* 
verbs and prepositions in Isizuln, enters in like manner into the 
composition of the same parts of speech in numerous African 
dialects, and is closely allied, if not identical* with a similar par- 
ticle in other families of language. Thus, in the Tete and Sena, 
pazuru, above ; panze or pandzej ;bfckxw ^pangorapangora, gradu* 
ally; pambari, by the side; paboze, only; pafupi^ near; pakuti, 
in the midst In the Sofala, padolco padako, gradually ; parAbe- 
dshe, before. In the Inhambane, papandshe, outside ; padokumia, 
slowly; pashani, above. In the Quilimane and Moeambiqne, 
changing p into another letter of the same orgaa, v; thus, vassuru 
and vazulu, above; vati^pansi in the Zulu, beneath; Mosam* 
bique, va, here, on this side; vafaariri, near; vamoza f once. In 
Cape Delgado, wakati ^ pakati of the Isizuin, in the midst ; 
papiri, sometimes. The Mpongire kas tw.= in, in the space of, 
both separable and in composition iwith other words, .especially 
those which denote time; thus, va, among, at; vate f sooh ; vote 
vena, now ; and the Benga hasptcfe, near. 

Not only as a prefix, and in general signification, but virtually 
in form also, this particle pa is found in the English prefix by or 
be, <xerman bei, Gothic hi, etc. And this prefix is perhaps allied 
to the Danish paa, and the Russian po; a the Latin has it in pos- 
sideo and a few other words;" and its prevailing sense and chief 
element are found again in the Sbemitic prefix b (a, beth) — a relic, 
perhaps, of an original language in common use before the dis- 
persion on the plain of Shinar, and a still living .ligament between 
the three divisions of the tripartite tongue since known as the 
Shemitic, Japhetic, and Hamitac.* 

The substance of this particle pa is seen also in the Zulu 
interrogative adverb pi, which is sometimes joined with the 
preposition nga; thus, ngapi naf where? whither? whenoe? close 
upon what ? about how many ?— and sometimes with the personal 
pronoun, the subject of inquiry; thus, upinat where is he? 
oapi na? zipina'f whereabout are they {in respect to situation, 
number, or quantity)? Nor is this use of the particle confined 
to the Isizulu. The Inhambane has tingapit how much ? Tete, 
bangapit how many? Mosambique, gavit and Cape Delgado, 
vingapit how many? Inhambane, tfaj» t where? whither? So- 

* We must be permitted to observe that, while the extensive analogies traced 
bj our correspondent among the language* of Africa appear to us highly interesting 
and important, we cannot regard it an safe, in the present state of phibogical sci- 
ence, to draw any inference from occasional and isolated resemblances to Indo-Eu- 
ropean or Semitic forms. When it shall be proved by cautious and comprehensive 
investigation, that a real connection exists between the Mo-European and Semitic 
families of languages, it will he tape to mate a similar attempt for o±ber .femjujet 
more widely and obviously diverse from each othcrv— Coopc. pr Pugf* 



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182 L. Grout, 

fala,api7 Tete and Sena, kuponit Quilimane, ion? Mosambiqtte, 
vai? Cape Delgado, depi? — where? whither? whence? So the 
Suaheli, wapi? the Nika, luapif the Hiau, kwapi? — where? 
whence? whither? Suaheli, wavgapi? which in number? In 
the Wakuafi we find the interrogative pronoun nit = what? and 
pa, combined in the word painiof why? whereby? wherefore? 
So in the Quilimane paranit why? because of. The Benga has 
ove? which? where? See also the Bornu language : yimpit at 
what time? when? ampi? which people? dandalpit which 
mosque? hirpit which slave? perpit which horse? 

The root of the same adverb is seen again in the Zulu inter- 
rogative po t poge f why ? po ini ? then why ? ini po ? why 
then ? and the classical scholar will readily observe the likeness, 
both in form and import, which this particle bears to the Greek 
TroD, where? mfc, how? not, whither? etc. 

Many Hebrew scholars derive the Hebrew preposition b (a) 
from the noun beth, house, in the house ; hence in, by, near. In 
some instances its Hamitic equivalent pa (po, va, vo) carries with 
it the idea of being at home ; thus, Sena, aripof is he at home ? 
Mosambique, ngi ya vo, I am at home ; mvkungu wa va, the 
master is at home ; u hi vo, he is absent. 

7. La is a demonstrative particle entering into the composi- 
tion of the demonstrative pronouns, and of a few of the adverbs ; 
thus, loku (la + vJcu), this, then, when ; lapa (la + apa [a +pa] ), 
here ; lapo, there ; tapaya, yonder. 

8. Ya is an adverbial suffix, derived, perhaps, from the verb 
vlcu ya, to go ; denoting distance in place, and generally accom- 
panied by some gesticulation, as pointing the finger, or inclining 
the head = yonder. Thus, leya, that, or there yonder ; lapaya, 
away yonder. The Suaheli has ya bide, far ; Pokomo, kuye, far, 
distant The Setshuana and Mpongwe make use of la in a 
similar manner : thus, Setshuana, fcakala, far, distant ; Mpongwe, 
la, distant : so the Hiau, kvla, distant 

9. Apa, apo, apaya. In the adverb apa, we have the insepa- 
rable pa = near, close, by, and the genitive particle a, which is 
sometimes preceded and strengthened bv the demonstrative la; 
thus, apa or lapa, here, at this place, hither, at the time, when ; 
apo or lapo, there, at that place, thither, where ; apaya or lapaya, 
yonder, at a distance. In some of the neighboring dialects, this 
adverb has reference to adjacent or contiguous time as well as 
place. Thus, Inhambane, apa, here, now ; apa apa, just now ; 
Mpongwe, vena, here ; vote, soon ; vate-vena, now ; vava, there ; 
vana and mevana, yonder; Suaheli, hapa, here; Nika, hiva; 
Pokomo, hafa; Hiau, hapano — here, hither; Suaheli, hapo, there; 
mahalihapa, hapano, thence; Nika, kuahiva, hence (from here); 
Benga, okava, here ; okavani, there ; ovani, there, yonder ; ove, 
where ; pam, this moment. 



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Particles of the Isizulu, etc. 183 

10. Kufupi = short, near, not far distant. The root of this 
word, fupi, may be traced in many cognates of the Isizulu : in 
the Inhambane, Tete and Sena, pafupi, near; Suaheli, mfopi, 
short; karubi, near; Nika, mfuhi, short; fefi, near; Kamba, mu- 
guwe, short; waguwe, near; Pokomo, mfufi, short ; hafufi, near; 
Emboma, Jcufe, short ; Mpongwe, pe and epe f short ; Setshuana, 
gaufi, near; Kongo, hofi, short. In both the form and import of 
this word, there is much to suggest that it may be radically a 
mere reduplication of pa, originally = papa, near by. 

11. Ezansi, pansi. The root nsi, or ami, which occurs in 
ezansi (ezi + ansini, contr. ezansi), the locative plural of an obso- 
lete Zulu noun izansi = sand, sea shore, bed of a river, and 
hence ezansi, signifying sea-ward, down country, lower down, 
aground — which root occurs also in pansi {pa + nsi or ansi) = 
aground, on the ground, down, beneath, below, under; and 
is doubtless seen also in amanzi = water, the sharp aspirate 
s having passed over into the weaker z of the same organ — 
is found, in substance, still in use with a similar meaning, in 
many Zulu cognates : Nhalemoe, nshi, sand ; Melon, nse, sand ; 
Ngoten, nshe; Mbofon and Udom, nshishe; Eafen, aseve; Orun- 
gu, deseye, pi. mosey e; Babuma, ndshie; TJndaza, eshei, pi. man- 
shk— sand ; Fanti, nsu, water ; Quilimane, nuinshi, river ; Zulu, 
amanzi, water, loc. emanzini, in the water ; Param, nzi and nze ; 
Papia, nshi and ndshi; Pati, ndsi; Bayon, ndshib ; Mbamba 
ana Bumbete, andsha and mandsha; Kiriman, mandshe — water. 
So also, Cape Delgado, madshi, water; pansi, low place; Tete 
and Sena, madzi, water; pandzi, low place; panze, or pandze, be- 
neath, on the ground ; Sofala, madshi, water ; pashi, low place ; 
Quilimane, (Kiriman?), mandshe or mainshe, water; Maravi, 
madze or matse, water; panze, beneath, on the ground; Cape 
Delgado, sini, beneath j Suaheli, madshi or madyi, water ; nti, 
earth; tini, below; tini ya-, under ; Nika, mazi, water ; st, earth; 
zini, below; ziniya-, under; Kamba, mansi, water; ndi, earth; 
deo, below; dec ya-, under; Pokomo, mazi, water; nsi, earth, 
below ; nsi yar, under ; Hiau, messi, water ; pasi ya-, under ; 
Setshuana, metse, water; tlase or thkue, below, beneath, under. 

The connection between ezansi and pansi in the Zulu dialect, 
and many of the above words in its cognates, will be more appar- 
ent by observing that the s in these Zulu words has a kind of 
guttural aspiration, which some have attempted to represent by 
the use of t, and by writing the words, as they are generally 
written in the Kafir (Xosa) dialect, thus, ezantsi, pantsi. 

12. Pezulu, pezu. The adverb pezulu, preposition pezu = over, 
above, on, upon {pa, near, at, + izvlu, sky, heaven), is found in 
many of the neighboring dialects. In some it consists of the 
noun alone ; in others, of the noun and preposition pa, or va, as 
in the Isizulu; thus, Tete and Sena, pazuru or huzuru, over, 



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184 L. Qrout, 

above, up, upwards; Mosambique, vazulu, utulu, ozulu, over, 
above ; Quilimane, vazuru, above ; Suaheli, ju, above ; ju ya*, 
over; Nika, zxdu, above; zuluya-, over; Kamba, ulu, above; zulu 
ya-, over ; Pokomo, zu, above ; ulu 10a-, over ; so the Isizulu, 
pezulu, above ; pezu kwa-, over. The noun itself may be traced 
much farther; the following are a few specimens of its forms in 
different dialects:. Fanti, . esuru, sky; Avekwom, ezvhe, sky; 
Kongo, ezulu; Eraboma, zulu; Basunde and Babuma, yulu; 
Mbomba and Bumbete, yob; Kabenda, yilu and Icuyilu; Kam- 
bali, usub emdozulo — sky, heaven. 

13. EnhlOy enhle, panhle. The preposition enhla, up, above ; 
and the adverb enhle, in the field, abroad, without ; also panhle 
(pa + enhk\ without, outside, abroad — are all derived from the 
noun inhla, an open field, waste, desert, wilderness, an unculti- 
vated, desolate section of countrv ; hence, an elevated, up-land 
district, since the natives prefer the rivers and fertile valleys; and 
hence the significations of the adverbs and preposition, abroad, 
without, above. The Inhambane has papandshe; Sofala, kuncU 
sha ; Tete and Sena, and Cape Delgado, kunsha and kundsha — 
without, outside ; Suaheli, nde; Nika and Pokomo, nse; Kamba, 
nsa ; and Setshuana, ka in tie — without, abroad. 

14. Pakati (pa + kati, the root of urnkati, space ; isUcati, time) 
= in the midst, between, within, inside. Among kindred dia- 
lects we have the following.: Delagoa, tshikarre ha-, in the midst 
of; Tete and Sena, mukaii, within ; pakati pa-, in the midst of; 
Gape Delgado, wakatiwa-, in the midst of; Suaheli, kati, middle; 
kati, katikoUi, between ; Nika, kahi, middle ; kahikahi, between ; 
Kamba, kati, middle ; kati ya-, between ; Pokomo, kaJii, middle ; 
kahi kahi, between ; Hiau, jirikati, middle ; pajirikati, between ; 
Mpongwe, gare, go gave, niiddle, centre, between ; Setshuana, ka 
gave, between. 

15. Kambi, kumbe, pambi, pambili, kahili. In the words Team* 
h\ of course ; kwmbe, perhaps ; pambi or parnbili, in front, be* 
fore ; and kahili, second (isibili, zimbili, etc., two), we find radi- 
cally the same element or elements, and the same generic idea, 
both in the Isizulu and in many of its kindred dialects, viz. : bi f 
mbi or mbe, bili, mbili, mbele = else, other, opposite ; and hence, 
second, two, in front, before, of course, perchance, perhaps. The 
root inbe is still heard occasionally, especially from the older 
men, as an adjective, in the sense of other, another ; thus, a ngi 
Vazi ilizwi dimbe, I do not know another saying, proverb ; so 
izindaba ezimbe, other matters (= izinduha ezniye). This root, mbi, 
having i final instead of *, is not uncommon in the Kafir (Xosa) 
dialect, where it also signifies another, other, a different one. 
In some cognates of the Isizulu we find one element of the full 
form mbili or vribdt, and in some another element; and in other 
cognates the two combined: thus, Pokomo, mbi, two; Ndob, be 



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Particles of the Isizulu, etc. 155 

and mbe; Kum, mbe and mba; Bagba, Bamozu, and Momenya, 
mbe ; Nhalemoe, Param, Papia, Pati, Musu, and Puka, mba — 
two; Kamba, M, two ; Kamoali, *'fe, two ; Suaheli, mbili ; Nika, 
nibiri: Kiriman, belt; Meto and Matatan, peU — two. Kamba, 
mbe; Suaheli, mbele; Nika, Pokomo, Hiau, and Cape Delgado, 
mbere — before ; Sena, kurnbare, opposite ; Tete, pambare, by the 
side; mbare, along beside; Sofida, pambedshe, before; Orungu, 
mbani, two; Mpong we, inborn vani, ambani, two; mbe, or; kambe 
and kambenle, wherefore; Benga, tombeti, either, or; ibali, two; 
Wakuafi, arte, warre, two ; Setshuana, gaberi, twice ; kapele, be- 
fore ; kampo, perhaps* So the verb, in Isizulu, pamba, cross, 
oppose; Mpongwe, simbia, oppose. We may notice also the 
resemblance, at least external, between some of these Hamitio 
words, as pambih\ kopek, before, in front, and the Hebrew k'bel 
(tap), the front, over against, before. 

1(5. Kade, hide. Corresponding to the root <fe (hade, hide, long, 
far), the Kamba has ndi, far; Suaheli, nde, abroad; Mj>ongwe, 
da, nda, long; Setshuana, gute, far; guteni, far off; Galla, dera, 
high ; Nika, hire, far ; Tete and Sena, fatten, far, distant. 

17. Katshana. According to the form of this word, we must 
regard it as a diminutive of kati (umkati, space) = a short space, 
a little distant, not far away. But the use of the word by the 
natives always indicates rather a longdistance, remote, faraway. 
Hence they sometimes define it by giving kade as a synonym ; 
and they have recommended it as a proper rendering of such 
phrases as the prayer " be not far from me, O Lord" = Unga bi 
katshana kumi, } Nkos\ It would be more in accordance with the 
signification which the natives give this word, to suppose it a 
diminutive of de, far, long, distant, a formation not much un- 
like impanjana (impandzhana), from impande; so kade, dim. &a* 
jana (kadzhana). The Efik has amjan, long; the Kongo, tshefa, 
long; Sofala, tambo, far, which would make the diminutive ten- 
jana (tandztiana), little far; Mandingo, jang, long. 

18. Eduze = near, close, not far away ; Galla, deo, near; Mo- 
sambique, uduli, after. 

19. Emva (emuva, ngemva, nga semva, Jcamva) = after, behind, 
in the Tear. The Tete and Sena dialects have buio, mumbuio, 
kumbuio, after ; ngambuire, beyond ; Benga, ombuwha, behind j 
Inhambane, muawe, behind. This word, emva, is a noun origin- 
ally, umva*, rear, from the verb va (uku va, to come) = come, 
follow after ; from which verb we have also the noun umvo, a 
remainder, or an excess over and above ten, twenty, thirty, or 
any exact number of tens — what comes after ten or tens. And, 
as the native counts with his fingers, when he has gone through 
with both hands and made up ten, he turns back = a buya, and 
goes over the same again. Further knowledge of kindred dia- 
lects may show still closer relation than we now see, between the 



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136 L. Grout, 

two verbs, uku va, to come, follow after, and uku buya, to return. 
Already we find : in the Batanga, via, come ; Mpongwe, bia ; 
Sofala, via; Mosambique, pia; Delagoa, buia— come; Nika, tua, 
follow; Suaheli, fiiata, follow; Nika, uya, return. The Mpongwe 
has the adverbs fa and va, again. 

20. Neno, nganeno. The radical substance and general import 
of the adverb and preposition neno and nganeno, signifying on 
this side, prevail extensively in the cognates of the Isizulu. In 
some dialects its use corresponds to that of the Zulu apa ; and in 
some instances we find the two, or parts of the two, combined 
in one word ; and in some dialects we find va, where others use 
either apa or neno. Thus, Hiau, hapano, here, hither; Suaheli, 
hapano, thence; Quilimane, uno; Tete and Sena, kuno — here, 
on this side ; Tete, zani kuno ; Maravi, dzani kuno— oome ye 
here; Mpongwe, gunu, here; Setshuana, monu, kuanu, here; 
kayenu, now, to-day; Quilimane, uvanene, now; Mosambique, 
nananu, nanano, now, soon, just now ; Tete, zapanupanu, now. 
In Isizulu and Inhambane we have apa; in Tete and Sena, kuno; 
and in Mosambique, va — here. 

This adverb and preposition neno is evidently compounded 
of a preposition (in tne Zulu, na), and the pronoun second per- 
son plural (in the Isizulu, the conjunctive, genitive form, inu or 
enu, the sharp final u being softened to o; thus na + inu or eno, 
= neno) = within from you, between the person speaking and 
those addressed, this side of, hither, here ; as nganeno kwako, this 
side of thee. Hence si lapa, we are here present {apa } close by) 
= si nenuy we are with you, on this side of (from which we aa- 
dress) vou. So in Tete and Sena, kuno (= Au, to, by, + no, soft- 
ened from the suffix pronoun nu [as in anu, wanu, zanu, etc. = 
yours, of you]) = by you; Suaheli, hapano (pa or apa, by, + the 
pronoun nui)\ Hiau, hapano — thence, hither, here; Setshuana, 
monu (wio, in, among, + enu, suffix pronoun second person plu- 
ral) = here. So kuanu (kua, at, + enu) = here ; Mpongwe, gunu 
(go, at, to, + anuwe, ye, contr. nu) = here. See also Tete, xapa- 
nupanu; Quilimane, uvanene; ana Mosambique, nananu, nanano 
— now, soon. 

21. Malungana (adverbial prefix ma, + lungana, be straight 
with — reciprocal form of the verb lunga, be straight, right) = 
straight with, over against, opposite to, side by side, near. 

22. Kodwa. The Isizulu has kodwa (ka + udwa or odwa) = 
only, simply, singly ; and its various pronominal forms, as nged- 
wa, sodwa, yedwa, bodwa, zodioa, etc., I, we, he, or they alone. 
The Inhambane has moido, muedo, one ; Quilimane, moda, modze, 
one ; Mosambique, moza ; Maravi, modze ; Kasands, Songo, and 
Kisama, moshi and most; Meto, modshi; Matatan, motsa and 
moza — one ; the Galla and Pokomo, koda, a part, portion ; the 
Galla, dua or dutva, empty, void, merely ; thus the Galla, ini 



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Particles qf the Isizulu, etc. 187 

harka duwa dufe 7 he comes with empty hands (i. e. hands alone, 
hands only) = Isizulu, izanhla zodwa. 

28. Kcenti (ka + nti or anti, the contrary, but) = on the con- 
trary, but, whereas, yet, nevertheless ; Setshuana, Jcanti, whilst ; 
Kamba, ndi, but, yet; Mpongwe, ndo, but; kande, because; 
Mandingo, tvarante, or, or else. Query — has this word any con- 
nection with the Greek d**/, Latin cmtet 
. 24. .iSe, vain, empty, naked; noun, ize, ilize 1 also ubuze, vanity, 
emptiness, nakedness, nothing; Sena, peze (pa + ize), false; pezi, 
in vain; zapezi, empty; Mpongwe, zyele, not, nothing. 

25. Kakulu (&a, of, + kulu, great ; verb, uku kuki, to grow 
large) = greatly. The root of this word is very common in the 
kindred dialects ; thus, in the Delagoa, the adjective kulu, great; 
Inhambane, hongoh; Sofala, guru ; Tete and Sena, kuru; Cape 
Delgado, kulu — great. So in the Nika, mkulu; Pokomo, mku; 
Kiriman, ula; Kisama, Lubalo, andLongo, kolu; Kasands, gola; 
Orungu, mpolo; Mpongwe, polu and mpolu — great; Setshuana, 
hagolu, greatly. 

26. Kutangi, day before yesterday; Suaheli, tangu, since; tangu 
miaka mrurilz, since two years; Nika and Pokomo, hangu, since ; 
hangu miaka miiri, since two years ; Pulo, hanki, yesterday. 

27. Izolo y yesterday; Delagoa, atolo; Sofala, Tete and Sena, 
xuru; Quilimane, nzura, nzib; Maravi, dzub; Nika and Poko- 
mo, zona; Kongo and Basunde^ zono; KirimaD, nzib; Nyombe, 
ckono; Mimboma, ozono; Musentandu, zonu; Ngoala, ezo — ^yes- 
terday, 

28. Kusasa (Aw, it, + sa, yet, + *a, dawns ; ekuseni, locative case 
of uku sa, to dawn) = early (to-morrow morning) ; Mosambique, 
viands early; utsha, utshaka, in the twilight; Kamba, katene, early. 

29. Bhnini, at mid-day, in the day-time ; Avekwom, emini, to- 
day; Efik, imjbn, to-day. 

80. Intamboma, 'inatambam\ afternoon, towards evening; Cape 
Delgado, rwremba, evening; Delagoa, adiamba va-pela, sunset; 
Inhambane, dambo ya gubele, sunset 

31. Ngomso (nga + umso, in the morning) = to-morrow, from 
the verb uku so, to dawn; Cape Delgado, matsesu] Pokomo, 
heso ; Suaheli, kesho ; Setshuana, usasane, — to-morrow. 

32. Namhla (na + umhla, with the day, this very day)=to-day ; 
Delagoa, namasha; Sofala, nyamashi; Inhambane, nyanse, to-day. 

S3* Ngemihla (nga + imihla, pi. of umhfa, day— by days)=daily ; 
Sena, tstko-zonke; Mpongwe, ntshug' wedu, (every day)— -daily. 

34. JEfodub, anciently ; mandulo, at first (from ukwandula, an- 
dulelaj to precede, be first); Mosambique, nyululu, old; Hiau, 
bngola; Nika, longola where ; Suaheli, tangulia mbde — precede. 

35. Kanye, knnye (io, Au, + wye, one)= once, at once, together ; 
Tete, kabosi; Sena, kabozi; Nika, vamenga; Kamba, wamue; Poko- 
mo and Mosambique, vamoza; Setshuana, gangue~-<mioe, together. 

VOL. VI. 18 



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1S8 L. Qrout, 

86. Kaningi, hminai (ha, hi, + ningi, much, many) = often, 
much, enough, plentifully; Delagoa, nyinge; Inhambane, singi, 
tingi; Tete and Sena, xinehe; Moeambique, indsfie; CapeDelga- 
do, nyingi— much ; Suaheli, Nika, and Pokomo, nery'i; Mpong- 
we, nyerwe; Setshuana, gantsi— much, often. 

87. iCaAfe, fatAfa (ia, fa^ + hJe, nice) = well, nicely, beauti- 
fully ; Nika, wizo; Kamba, neza; Setsnuana, eingthm— well. 

88. -?Wi, again, often; Sena,,/Wt, since; Mpongwe, fa, again. 
Compare Gothic, vfta; Engliph and Saxon, oft, often, etc. 

89. Nxa (noun inoca = side, sake, portion, interest) = where, 
if, when. Ngenwa (nga 4- tnao), on account of 

40. JE>, a word, or part of a word, probably from the verb ha; 
usually classed as an adverb, and used sometimes by itself, espe- 
cially in a negative connection, but more frequently in composi- 
tion, to signify present, extant, in being, here, there. The Tete 
and Sena dialects have vho, there ; Suaheli and Pokomo, huho ; 
Nika, hxho; Hiau, akoho — there; Setehuana, mo, hua; and 
Mpongwe, gogo — there. 

41. Konje, maty* = immediately, now, speedily, are generally 
supposed to be compounded of the adverbial ©reformative ho or 
ma, and nje « thus, so, in like manner. But the ordinary use of 
nie, nja, does not readily suggest the idea generally expressed by 
tnese words honje, manje = immediately, etc., unless we are to 
suppose that the notion of similarity, which nja is used to ex- 
press, bears hard upon the notion of sameness = same time, at 
onoe — a suggestion which has some color of support from the 
use of the probable synonym ga in the Mpongwe dialect, which 
is there defined as signifying both like and same. For further 
remarks on these words, see the next. 

42. Masinyane (masinya, hamsinyane, hamsinya) = soon, im- 
mediately, speedily, quick, now, was probably derived from some 
noun or verb (now obsolete in the IsizuluV signifying speed, 
to hasten, be quick. And keeping in mini the laws of muta- 
tion among consonants in the Isizulu and its cognates — that s 
sometimes gives place to to ; that b changes to tsh, and sometimes 
toy; m to ny; and mb, to w/— bearing in mind also that the 
nasal m orn is not really radical in some words, but introduced 
to soften down the hard elastic nature of a mute, m being taken 
by a labial, and n by a lingual — it is not improbable that farther 
researches may prove both manje and masinyane, and possibly 
the verb tshetsha, to have a common origin, and to be, perhaps, 
radically the same as some of the following words in cognate 
dialects: Sofala, Nika, and Pokomo, sambi, now; Cape Delgado, 
sambe (changing mb to nj = sanje, meaning the same as the Zulu 
tn<mjfe)«now; Hiau, sambano; Tete and Sena, hutshimbetsa — 
quick; tektmbitsa, shimbi&a, shimbisisa, tshimbiza, fast, quickly; 
ku tshmbi'tshdmbi Lshambizwo, immediately, soon. 



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Particles of the Isizulu, etc. 1S9 

Now take One of these words, the verb shirnbisa, change the 
causative into the reciprocal form, shimbana, restore the radical 
consonants to their original strength and simplicity, sh to 5, and 
fob to i, and we have vibana, the form of which in the locative 
would be sinyaneni, contr. sinyane; and prefixing the adverbial 
incipient ma, we have masinyane; and from this, by prefixing ia, 
we have kama&inyane, contr. kamsvfiyane; contr. again, kamsinya. 

43. Ngesflwnu, ngamabomu (nga + isibomu or amabomu, pur- 
pose, design) =iby design, on purpose, willfUlly; Nika, mbomu, 
great; ubomu, greatness. 

44. Nja, nje y an adverbial particle, signifying like, as, so, thus, 
and used, for the most part, in composition ; thus, njaio, mam, 
njenga, konje. Kamba, jauf how? Hiau, hiajiiit how? Man- 
dingo, nya, a manner, a method; nyadit howr in what way? 
Mpongwe, ga, like, same; egaleni, like, similar; Benga, njat 
who? what? Setshuana, yaka r yualeka, as, like*, yuana or yana % 
yiuilo or yah, thus. 

45. Njafo (nja, like, + fo, dem. adv., this) = like this, so, thus, 
likewise. Kanjah (ka + njalo), thus, so, likewise. Inhambane, 
kararo, thus ; Galla, alcana, thus ; Mpongwe, ga, egaleni, yena, 
nana, and ka, so, thus, after this fashion ; Setshuana, yudb, yah, 
thus. 

46. Init (i r it, + mt what? yim r euphonic y + ini; yini naT) 
= why? whether? Inhambane, para Icinmit why? Quilimane, 
parani t why? CapeDelgado, ninit why? Suaheli, Nika, and 
rokomo, kuanit why? 

47. Nganinat (nga, with or by, + nit what?) =» how? why? 
wherefore? Inhambane, para kinanif why? So&la, ngenyef 
why? Quilimane, parani t why? Suaheli, kuanit kua ninit 
ganit Nika and Pokomo, kuanit ninit — why? wherefore? 

48. Npani nat kanjani nat (ka + nja + mf)=» how? like 
what? Inhambane, karinit how? Sena, kutamf Suaheli, Nika, 
and Pokomo, kuanit ninit — how? 

49. Nini nat (nit what? + nit what?) = when? Sofala, 
Tete and Sena, Nika, and Hiau, rinit Quilimane and Mosam- 
bique, Knit — when? 

50. Ai, the negative a prolonged and strengthened by the aid 
of the vowel i, and sometimes also by the semivowel y, ayi ; or 
it may take also an initial breathing h, giving Aai, or hayi — no. 
The Mosambique has vai; Maravi, iai; Tete, di-ai ; Mandingo, 
ara; Setshuana, ga — no. 

61. AUshe (aiy no, + tshe f no, obsolete in the Isizulu, but still 
in use among the Betshuana) = no, not so, not that, but; Sua- 
heli, sifio; Nika,. sefio — no. 

52. Amanga T a noun phrral, signifying deception, falsehood, 
pretense ; hence the adverbial meaning, no, not so, it is false — 
from the verb uku unga, to feign, deceive, entice. This root 



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140 L. Grout, Particles of the Isizulu, etc. 

unga si still found all along the eastern coast of Africa, in both 
nouns and verbs, all of the same import as amanga and uhi 
wrvga in the Isizulu ; thus, Cape Delgado, ulcmgo, it is false, a 
falsehood, a lie ; si ulongo, it is not a he ; Suaheli, urongo, a lie ; 
nena urongo, to tell a lie ; Nika, ukmgo, a lie, to tell a lie ; Kam- 
ba, uwwvgu, a lie; ajia uvrungu, to tell a lie; Pokomo, muongo, 
a lie ; Hiau, anga, a lie ; Mpongwe, noka; Setshuana, aka — to lie. 

53. Ewe, a simple form of assent = yes; Suaheli, eiwa; Kam- 
bo, uo, wiu — yes. 

54. Ehe, an expression of assent = yes, it is so. Mpongwe, 
ih; Mandingo, aha; Benga and Setshuana, e or eh — yes. 

55. Itshi, itshilo — the first form a contraction of the second 
(the pronoun t, referring to inkosi, the chief, + tshilo, the present 
perfect tense of tsho, speak) = he has spoken, assented, affirmed; 
hence, yes, truly, it must be so. 

56. Yebo (ye + bo) = yes, indeed I Setshuana, ebo; Mandingo, 
yei— yes. 

In the Isizulu, as in many other languages, especially among 
the tribes of Africa, the same word appeaTs, according to its use 
and connection, sometimes as an adverb, and sometimes as a 
preposition, or as a conjunction. Several words which are used 
in the twofold capacity of an adverb and a preposition, when 
they serve as the latter, are alwavs followed oy another, as by 
hwa or na; thus, pezu Jewomuti (fam + umvti), upon the tree; 
edvze nentaba (na + intaba), near the mountain. 

This use of a complements! preposition prevails in many of 
the cognates of the Isizulu ; thus, in the Tete and Sena, pakati 
pa- % in the midst of; as pakati patsika, in, the midst of the night, 
midnight ; Inhambane, bakari nya- ; as bakari nyaushtgu, in the 
midst of the night, midnight; Delagoa, tshikare kadiambo, mid- 
day; Mosambique, nzua va-\ as nzua vamuru, midday; Cape 
Delgado, waJcati wa- ; as wakati wamfula, in the midst of the 
rainy season, winter. So in the Suaheli, tint ya-, under ; ju ya- y 
over ; Nika, zini ya* } under ; zulu ya-, over ; Pokomo, nsi ya- } 
under ; ulu wa- y over ; Hiau, pasi ya- f under, etc., like the Zulu 
pansi kwa-, under, pezu kwa-, over. Or perhaps these and similar 
examples should oe considered as instances of prepositions fol- 
lowed by the genitive, and more like the use of ngenxa ya- ; 
thus, ngenxa ya&e, on account of him, for cause of him. In fact, 
all examples of this kind serve to confirm the opinion that many 
of the prepositions were originally nouns. 

Umsunduzi, May 8th, 1858. 



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ARTICLE III. 



TRANSLATION 

C» IBS 

SURYA-SIDDHANTA, 

A TEXT-BOOK OF HINDU ASTRONOMY; 

WITH NOTES, AND AN APPENDIX. 
Br R«v. EBENEZER BURGESS, 

fOBMEBLT MSOONAEY OF THX A, B. 0. F. X. »T IFDIA, 
A8BI8TSD BY TBI OOMMITTSB OF PUBLICATION. 



Presented to the Society May 17, 1858. 



Introductory Note. 

Soon after my entrance upon the missionary field, in the Maratha 
country of western India, in the year 1839, my attention was directed 
to the preparation, in the Marathi language, of an astronomical text- 
book for schools. I was thus led to a study of the Hindu science of 
astronomy, as exhibited in the native text-books, and to an examination 
of what had been written respecting it by European scholars. I at 
once found myself, on the one hand, highly interested by the subject 
itself, and, on the other, somewhat embarrassed for want of a satisfactory 
introduction to it. A comprehensive exhibition of the Hindu system had 
nowhere been made. The Astronomie Indienne of Bailly, the first ex- 
tended work upon its subject, had long been acknowledged to be founded 
upon insufficient data, to contain a greatly exaggerated estimate of the 
antiquity and value of the Hindu astronomy, and to have been written 
for the purpose of supporting an untenable theory. The articles in the 
Asiatic Researches, by Davis, Colebrooke, and Bentley, which were the 
first, as they still remain the most important, sources of knowledge re- 
specting the matters with which they deal, relate only to particular 
points in the system, of especial prominence and interest JBentley's 
volume on Hindu astronomy is mainly occupied with an endeavor to 
ascertain the age of the principal astronomical treatises, and the epochs 
of astronomical discovery and progress, and is, moreover, even in these 
respects, an exceedingly unsafe guide. The treatment of the subject by 
Delambre, in his History of Ancient Astronomy, being founded only 



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142 E. Burgess, etc., 

upon Bailly and the earliest of the essays in the Asiatic Researches, 
partakes, of course, of the incompleteness of his authorities. Works 
of value have been published in India also, into which more or less of 
Hindu astronomy enters, as Warren's Eala Sankalita, Jervis's Weights 
Measures and Coins of India, Hoisington's Oriental Astronomer, and 
the like ; but these, too, give, for the most part, hardly more than the 
practical processes employed in parts of the system, and they are, like 
many of the authorities already mentioned, only with difficulty accessi- 
ble. In short, there was nothing in existence which showed the world 
how much and how little the Hindus know of astronomy, as also their 
mode of presenting the subject in its totality, the intermixture in their 
science of old ideas with new, of astronomy with astrology, of observa- 
tion and mathematical deduction with arbitrary theory, mythology, 
cosmogony, and pure imagination. It seemed to me that nothing would 
so well supply the deficiency as the translation and detailed explication 
of a complete treatise of Hindu astronomy : and this work I accord- 
ingly undertook to execute. 

Among the different Siddhantas, or text-books of astronomy, existing 
in India in the Sanskrit language, none appeared better suited to my 
purpose than the Surya-Siddhanta. That it is one of the most highly 
esteemed, best known, and most frequently employed, of all, must be 
evident to any one who has noticed how much oftener than any other 
it is referred to as authority in the various papers on the Hindu astron- 
omy. In fact, the science as practised in modern India is in the greater 
part founded upon its data and processes. In the lists of Siddhantas 

g'ven by native authorities it is almost invariably mentioned second, the 
rahma-Siddhanta being placed first : the latter enjoys this prominence, 
perhaps, mainly on account of its name ; it is, at any rate, compara- 
tively rare and little known. For completeness, simplicity, and concise- 
ness combined, the Surya-Siddhanta is believed not to be surpassed by 
any other. It is also more easily obtainable. In general, it is difficult, 
without official influence or exorbitant pay, to gam possession of texts 
which are rare and held in high esteem. During my stay in India, I 
was able to procure copies of only three astronomical treatises besides 
the Surya-Siddhanta ; the ^akalya-Sanhita of the Brahma-Siddhanta, 
the Siddhanta-Qiromani of Bhaskara, and the Graha-Laghava, of which 
the two latter have also been printed at Calcutta. Of the Surya- 
Siddhanta I obtained three copies, two of them giving the text alone, 
and the third also the commentary entitled Gudharthaprakacaka, by 
Ranganatha, of which the date is unknown to me. The latter manu- 
script agrees in all respects with the edition of the Surya-Siddhanta, 
accompanied by the same commentary, of which the publication, in the 
series entitled Bibliotheca Indies, has been commenced in India by 
an American scholar, and a member of this Society, Prof* Fitz-Edward 
Hall of Benares ; to this I have also had access, although not until my 
work was nearly completed. 

My first rough draft of the translation and notes was made while I 
was still in India, with the aid of Brahmans who were familiar with the 
Sanskrit and well versed in Hindu astronomical science. In a few points 
also I received help from the native Professor of Mathematics in the 



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Sdrya-Siddhdnta. 143 

Sanskrit College at Puna. But notwithstanding this, there remained 
not a few obscure and difficult points, connected with the demonstration 
and application of the processes taught in the text In the solution of 
these, I have received very important assistance from the Committee of 
Publication of the Society. They have also — the main share of the 
work falling to Prof. Whitney — enriched the notes with much additional 
matter of value. My whole collected material, in fact, was placed in 
their hands for revision, expansion, and reduction to the form best 
answering to the requirements of modern scholars, my own engrossing 
occupations, and distance from the place of publication, as well as my 
confidence in their ability and judgment, leading me to prefer to intrust 
this work to them rather than to undertake its execution myself. 

We have also to express our acknowledgments to Mr. Hubert A. 
Newton, Professor of Mathematics in Yale College, for valuable aid ren- 
dered us in the more difficult demonstrations, and in the comparison of 
the Hindu and Greek astronomies, as well as for his constant advice and 
suggestions, which add not a little to the value of the work. 

The Surya-Siddhanta, like the larger portion of the Sanskrit litera- 
ture, is written in the verse commonly called the floka, or in stanzas of 
two lines, each line being composed of two halves, or pddas, of eight 
syllables each. With its metrical form are connected one or two pecu- 
liarities which call for notice. In the first place, for the terms used 
there are often many synonyms, which are employed according to the 
exigencies of the verse : thus, the sun has twelve different names, Mars 
six, the divisions of time two or three each, radius six or eight, and so 
on. Again, the method of expressing numbers, large or small, is by 
naming the figures which compose them, beginning with the last and 

foing backward ; using for each figure not only its own proper name, 
ut that of any object associated in the Hindu mind with tne number it 
represents. Thus, the number 1,5 77,9 17,828 (i. 37) is thus given: 
Vasu (a class of deities, eight in number) -two-eight-mountain (the seven 
mythical chains of mountains) -form-figure (the nine digits) -seven-moun- 
tain4unar days (of which there are fifteen in the half-month). Once 
more, the style of expression of the treatise is, in general, excessively 
concise and elliptical, often to a degree that would make its meaning 
entirely unintelligible without a commentary, the exposition of a native 
teacher, or such a knowledge of the subject treated of as should show 
what the text must be meant to say. Some striking instances are 
pointed out in the notes. This over-conciseness, however, is not wholly 
due to the metrical form of the treatise : it is characteristic of much of 
the Hindu scientific literature, in its various branches ; its text-books are 
wont to be intended as only the text for written comment or oral expli- 
cation, and hint, rather than fully express, the meaning they contain. 
In our translation, we have not thought it worth while to indicate, by 
parentheses or otherwise, the words and phrases introduced by us to 
make the meaning of the text evident : such a course would occasion 
the reader much more embarrassment than satisfaction. Our endeavor 
is, in all cases, to hit the true mean between unintelligibility and diffuse- 
ness, altering the phraseology and construction of the original only so 



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144 E. Burgess, etc., 

far as is necessary. In both the translation and the notes, moreover, 
we keep steadily in view the interests of the two classes of readers for 
whose benefit the work is undertaken : those who are orientalists with- 
out being astronomers, and those who are astronomers without being 
orientalists. For the sake of the former, our explanations and demon- 
strations are made more elementary and full than would be necessary, 
were we addressing mathematicians only : for the sake of the latter, we 
cast the whole into a form as occidental as may be, translating every 
technical term which admits of translation : since to compel all those who 
may desire to inform themselves respecting the scientific content of the 
Hindu astronomy to learn the Sanskrit technical language would be 
highly unreasonable. To furnish no ground of complaint, however, to 
those who are familiar with and attached to these terms, we insert them 
liberally in the translation, in connection with their English equivalents. 
The derivation and literal signification of the greater part of the tech- 
nical terms employed in the treatise are also given in the notes, since 
such an explanation of the history of a term is often essential to its 
full comprehension, and throws valuable light upon the conceptions of 
those by whom it was originally applied. 

We adopt, as the text of our translation, the published edition of the 
Siddhanta, referred to above, following its readings and its order of ar- 
rangement, wherever they differ, as they do in many places, from those 
of the manuscripts without commentary in our possession. The dis- 
cordances of the two versions, when they are of sufficient consequence 
to be worth notice, are mentioned in the notes. 

As regards the transcription of Sanskrit words in Roman letters, we 
need only specify that c represents the sound of the English ch in 
" church," Italian c before e and i : that,; is the English j : that f is pro- 
nounced like the English sh, German sck, French ch, while *h is a sound 
nearly resembling it, but uttered with the tip of the tongue turned back 
into the top of the mouth, as are the other lingual letters, £, d, n : 
finally, that the Sanskrit r used as a vowel (which value it has also in 
some of the Slavonic dialects) is written with a dot underneath, as r. 

The demonstrations of principles and processes given by the native 
commentary are made without the help of figures. The figures which 
we introduce are for the most part our own, although a few of them 
were suggested by those of a set obtained in India, from native mathe- 
maticians. 

For the discussion of such general questions relating to this Siddhanta 
as its age, its authorship, the alterations which it may have undergone 
before being brought into its present form, the stage which it represents 
in the progress of Hindu mathematical science, the extent and character 
of the mathematical and astronomical knowledge displayed in it, and 
the relation of the same to that of other ancient nations, especially of 
the Greeks, the reader is referred to the notes upon the text. The form 
in which our publication is made does not allow us to sum up here, in 
a preface, the final results of our investigations into these and kindred 
topics. It may perhaps be found advisable to present such a summary 
at the end of the article, in connection with the additional notes and 
other matters to he. there given. 



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SftRYA-SIDDHANTA. 



CHAPTER L 

OF THE MEAN MOTIONS OF THE PLANETS. 

Contents : — 1, homage to the Deity ; 2-9, revelation of the present treatise ; 10-11, 
modes of dividing time ; 11-12, subdivisions of n day; 12-14, of a year ; 14-17, 
of the Ages; 18-19, of an uEon; 20-21, of Brahma's life; 21-23, part of it 
already elapsed ; 24, time occupied in the work of creation ; 26-27, general 
account of the movements of the planets ; 28, subdivisions of the circle; 29-83, 
number of revolutions of the planets, and of the moon's apsis and node, in an 
Age; 84-39, number of days and months, of different kinds, in an Age ; 40, in an 
JSon ; 41-44, number of revolutions, in an JEon, of the apsides and nodes of the 
planets ; 45-47, time elapsed from the end of creation to that of the Golden Age ; 
48-61, rule for the reduction to civil days of the whole time since the creation ; 
51-62, method of finding the lords of the day, the month, and the year ; 53-54, 
rule for finding the mean place of a planet, and of its apsis and node ; 65, to find 
the current year of the cycle of Jupiter ; 56, simplification of the above calcula- 
tions; 67-58, situation of the planets, and of the moon's apsis and node, at the 
end of the Golden Age; 59-60, dimensions of the earth; 60-61, correction, for 
difference of longitude, of the mean place of a planet as found ; 62, situation of 
the principal meridian ; 68-66, ascertainment of difference of longitude by differ- 
ence between observed and computed time of a lunar eclipse ; 66, difference of 
time owing to difference of longitude ; 67, to find the mean place of a planet for 
any required hour of the day ; 68-70, inclination of the orbits of the planets. 

1. To him whose shape is inconceivable and unnianifested, 
who is unaffected by the qualities, whose nature is quality, 
whose form is the support of the entire creation — to Branma be 
homage ! 

The usual propitiatory expression of homage to some deity, with 
which Hindu works are wont to commence. 

2. When but little of the Golden Age Qcrta yuga) was left, a 
great demon (asura), named Maya, being desirous to know that 
mysterious, supreme, pure, and exalted science, 

3. That chief auxiliary of the scripture (veddnga), in its en- 
tirety — the cause, namely, of the motion of the heavenly bodies 
(jyotis), performed, in propitiation of the Sun, very severe re- 
ligious austerities. 

vol. vi. 19 



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146 E. Barges*, etc.. [i. 3- 

According to this, the Surya-Siddhanta was revealed more than 
2,164,060 years ago, that amount of time having elapsed, according to 
Hindu reckoning, since the end of the Golden Age ; see below, under 
verse 48, for the computation of the period. As regards the actual 
date of the treatise, it is, like all dates in Hindu history and the history 
of Hindu literature, exceedingly difficult to ascertain. It is the more 
difficult, because, unlike most, or all, of the astronomical treatises, the 
Surya-Siddhanta attaches itself to the name of no individual as its 
author, but professes to be a direct revelation from the Sun (surya). A 
treatise of this name, however, is confessedly among the earliest text- 
books of the Indian science. It was one of the five earlier works upon 
which was founded the Pauca-siddhantika, Compendium of Five As- 
tronomies, of Varaha-mihira, one of the earliest astronomers whose works 
have been, in part, preserved to us, and who is supposed to have lived 
about the beginning of the sixth century of our era. A Surya-Siddhanta 
is also referred to by Brahmagupta, who is assigned to the close of the 
same century and the commencement of the one following. The argu- 
ments by which Mr. Bentley (Hindu Astronomy, p. 158, etc.) attempts 
to prove Varaha-mihira to have lived in the sixteenth century, and his 
professed works to be forgeries and impositions, are sufficiently refuted 
by the testimony of al-Biruni (the same person as the Abu-r-Uai}ian, so 
often quoted in the first article of this volume), who visited India under 
Mahmud of Ghazna, and wrote in A.D. 1031 an account of the coun- 
try : he speaks of Varaha-mihira and of his Panca-siddhantika, assign- 
ing to both nearly the same age as is attributed to them by the modern 
Hindus (see Reinaud in the Journal Asiatique for Sept.-Oct. 1844, iv me 
Serie, iv. 286; and also his Memoire sur l'lnde). He also speaks of the 
Surya-Siddhanta itself, and ascribes its authorsnip to Lata (Memoire sur 
Tlnde, pp. 331, 332), whom Weber (Vorlesungen uber Indische Litera- 
ti! rgeschichte, p. 229) conjecturally identifies with a Ladha who is cited 
by Brahmagupta. Bentley has endeavored to show by internal evi- 
dence that the Surya-Siddhanta belongs to the end of the eleventh 
century : see below, under verses 29-34, where his method and results 
are explained, and their value, estimated. 

Of the six Vedangas, " limbs of the Veda," sciences auxiliary to the 
sacred scriptures, astronomy is claimed to be the first and chie^ as rep- 
resenting the eyes ; grammar being the mouth, ceremonial the hands, 
prosody the feet, etc. (see Siddhanta-^iromani, i. 12-14)* The import- 
ance of astronomy to the system of religious observance lies in the fact 
that by it are determined the proper times of sacrifice and the like. 
There is a special treatise, the Jyotisha of Lagadha, or Lagata, which, 
attaching itself to the Vedic texts, and representing a more primitive 
phase of Hindu science, claims to be the astronomical Vedanga ; but it 
is said to be of late date and of small importance. 

The word jyotis, " heavenly body," literally " light," although the 
current names for astronomy and astronomers are derived from it, does 
not elsewhere occur in this treatise. 

4. Gratified by these austerities, and rendered propitious, the 
Sun himself delivered unto that Maya, who besought a boon, 
the system of the planets. 



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i.'«.] . S&ryaSiddJidnta. 147 

The blessed Sun spoke : 

5. Thine intent is Known to me; I am gratified by thine aus- 
terities ; I will give thee the science upon which time is founded, 
the grand system of the planets. 

6. No one is able to endure my brilliancy ; for communication 
I have no leisure; this person, who is a part of me, shall relate 
to thee the whole. 

The manuscripts without commentary insert here the following verse : 
u Go therefore to Romaka-city, thine own residence ; there, under- 
going incarnation as a barbarian, owing to a curse of Brahma, I will 
impart to thee this science." 

If this verse really formed a part of the text, it would be as clear 
an acknowledgment as the author could well convey indirectly, that 
the science displayed in his treatise was derived from the Greeks. 
Romaka-city is Rome, the great metropolis of the West ; its situation is 

fiven in a following chapter (see xii. 39) as upon the equator, ninety 
egrees to the west of India. The incarnation of the sun there as a 
barbarian, for the purpose of revealing astronomy to a demon of the 
Hindu Pantheon, is but a transparent artifice for referring the foreign 
science, after all, to a Hindu origin. But the verse is clearly out of 
place here; it is inconsistent with the other verses among which it 
occurs, which give a different version of the method of revelation. 
How comes it here then ? It can hardly have been gratuitously devised 
and introduced. The verse itself is found in many of the manuscripts 
of this Siddhanta ; and the incarnation of the Sun at Romaka-city, 
among the Yavanas, or Greeks, and his revelation of the science of 
astronomy there, are variously alluded to in later works ; as, for instance, 
in the Jfiana-bhaskara (see Weber's Catalogue of the Berlin Sanskrit 
Manuscripts, p. 287, etc.), where he is asserted to have revealed also the 
Romaka-Siddhanta. Is this verse, then, a fragment of a different, and 
perhaps more ancient, account of the origin of the treatise, for which, 
as conveying too ingenuous a confession of the source of the Hindu 
astronomy, another has been substituted later? Such a supposition, 
certainly, does not lack plausibility. There is something which looks 
the same way in the selection of a demon, an Asura, to be the medium 
of the sun's revelation ; as if, while the essential truth and value of the 
system was acknowledged, it were sought to affix a stigma to the source 
whence the Hindus derived it. Weber (Ind. Stud. ii. 243 ; Ind. Lit. p. 
225), noticing that the name of the Egyptian sovereign Ptolemaios 
occurs in Indian inscriptions in the form Turamaya, conjectures that 
Asura Maya is an alteration of that name, and that the demon Maya ac- 
cordingly represents the author of the Almagest himself; and the conjec- 
ture is powerfully supported by the fact that al-Blruni (sec Rcinaud, as 
above) ascribes the Paulica-Siddhanta, which the later Hindus attribute 
to aPuli^a, to Paulus al-Yunanl, Paulus the Greek, and that another of the 
astronomical treatises, alluded to above, is called the Romaka-Siddhanta. 
It would be premature to discuss here the relation of the Hindu 
astronomy to the Greek ; we propose to sum up, at the end of this 
work, the evidence upon the subject which it contains. 



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148 M Burgess, etc., [i. 7- 

7. Thus having spoken, the god disappeared, having given 
directions unto the part of himself. This latter person thus ad- 
dressed Maya, as he stood bowed forward, his hands suppliantly 
joined before him : 

8. Listen with concentrated attention to the ancient and exalted 
science, which has been spoken, in each successive Age, to the 
Great Sages (maharshi), by the Sun himself. 

9. This is that very same original text-book which the Sun of 
old promulgated : only, by reason of the revolution of the Ages, 
there is here a difference of times. 

According to the commentary, the meaning of these last verses is 
that, in the successive Great Ages, or periods of 4,320,000 years (see 
below, under vv. 15-17), there are slight differences in the motions of 
the heavenly bodies, which render necessary a new revelation from time 
to time on the part of the Sun, suited to the altered conditions of things ; 
and that when, moreover, even during the continuance of the same Age, 
differences of motion are noticed, owing to a difference of period, it is 
customary to apply to the data given a correction, which is called btja. 
All this is very suitable for the commentator to say, but it seems not a 
little curious to find the Sun's superhuman representative himself in- 
sisting that this his revelation is the same one as had formerly been 
made by the Sun, only with different data. We cannot help suspecting 
in the ninth verse, rather, a virtual confession on the part of the promul- 
gators of this treatise, that there was another, or that there were others, 
m existence, claiming to be the sun's revelation, or else that the data 
presented in this were different from those which had been previously 
current as revealed by the Sun. We shall have more to say hereafter 
(see below, under w, 29-34) of the probable existence of more than 
one version of the Surya-Siddha-nta, of the correction called bijdy and 
of its incorporation into the text of the treatise itself. The repeated 
revelation of the system in each successive Great Age, as stated in verse 
8, presents no difficulty. It is the Puranic doctrine (see Wilson's Vishnu 
Purana, p. 269, etc.) that during the Iron Age the sources of knowledge 
become either corrupted or lost, so that a new revelation of scripture, 
law, and science becomes necessary during the Age succeeding. 

10. Time is the destroyer of the worlds ; another Time has 
for its nature to bring to pass. This latter, according as it is 
gross or minute, is called by two names, real {mUrta) and unreal 
{amiirtcCy 

There is in this verse a curious mingling together of the poetical, the 
theoretical, and the practical. To the Hindus, as to us, Time is, in a 
metaphorical sense, the great destroyer of all things; as such, he is 
identified with Death, and with Yama, the ruler of the dead. Time, 
again, in the ordinary acceptation of the word, has both its imaginary, 
and its appreciable and practically useful divisions : the former are called 
real (murta, literally " embodied "), the latter unreal (am&rta, literally 
44 unembodied"). The following verse explains these divisions more fully. 



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1 12.] Stirya-Siddhdnta. 148 

The epithet kalanAtmaka, applied to actual time in the first half of 
the verse, is not easy of interpretation. The commentary translates it 
44 is an object of knowledge, is capable of being known," which does not 
seem satisfactory. It evidently contains a suggested etymology (k&la, 
44 time," from kalana) y and in translating it as above we have seen in it 
also an antithesis to the epithet bestowed upon Time the divinity. 
Perhaps it should be rather " has for its office enumeration " 

11. That which begins with respirations (prdna) is called real ; 
that which begins with atoms (trutj) is called unreal. Six respi- 
rations make a vinddi, sixty of these a nddi ; 

12. And sixty nadis mate a sidereal day and night. . . . 

The manuscripts without commentary insert, as the first half of v. 11, 
the usual definition of the length of a respiration : " the time occupied 
in pronouncing ten long syllables is called a respiration." 

The table of the divisions of sidereal time is then as follows : 

10 long syllables (gvrvakthara) = i respiration {prdna, period of four seconds) ; 
6 respirations = i vioadl (period of twenty-four seconds) ; 

' 6o vin&dis = i nadi (period of twenty -four minutes); 

6o nadis = i day. 

This is the method of division usually adopted in the astronomical 
text-books : it possesses the convenient property that its lowest sub- 
division, the respiration, is the same part of the day as the minute is of 
the circle, so that a respiration of time is equivalent to a minute of 
revolution of the heavenly bodies about the earth. The respiration is 
much more frequently called asu, in the text both of this and of the 
other Siddhantas. The vinadt is practically of small consequence, and 
is only two or three times made use of in the treatise : its usual modern 
name is paid, but as this term nowhere occurs in our text, we have not 
felt justified in substituting it for vinadt. For nadi also, the more 
common name is danda, but this, too, the Surya-Siddhanta nowhere 
employs, although it uses instead of nadi, and quite as often, nddikd and 
ghatUcd. We shall uniformly make use in our translation of the terms 
presented above, since there are no English equivalents which admit of 
oeing substituted for them. 

The ordinary Puranic division of the day is slightly different from the 
astronomical, viz : 

1 5 twinklings (nimesha) = i bit (kdshthd); 

3o bits = x minute (kald) ; 

3o minutes = i hour {muhurta) ; 

3o hours = i day. 

Manu (i. 64) gives the same, excepting that he makes the bit to con- 
sist of IS twinklings. Other authorities assign different values to the 
lesser measures of time, but all agree in the main fact of the division of 
the day into thirty hours, which, being perhaps an imitation of the 
division of the month into thirty days, is unquestionably the ancient and 
original Hindu method of reckoning time. 

The Surya-Siddhanta, with commendable moderation, refrains from 
giving the imaginary subdivisions of the respiration which make up 



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150 E. Burgess, etc., [1 12- 

" unreal " time. They are thus stated in Bh&skara's Siddhanta-Qiromani 
(i. 19, 20), along with the other, the astronomical, table : 

ioo atoms (truti) = i speck (tatpard) ; 

3o specks = i twinkling (nimeska) ; 

18 twinklings = i bit (kdtkthd) ; 

3o bits = i minute (hold) ; 

3o minutes = i half-hour {ghatikd) ; 

2 half-hours = i hour (kthana) ; * < 

3o hours = l day. 

This makes the atom equal to ^ 81B rfjoo Tnr^ °f a d ftV > or TnrV&tjtk 
of a seeond. Some of the Pnranas (see Wilson's Vish. Pun p. 22) give 
a different division, which makes the atom about -rrVir^ of a second ; 
but they carry the division three steps farther, to the subtilissima 
(param&nu), which equals a jjbo joo o tnr tn °f a day, or ver 7 nearly 
•trHrtrTr^ 1 °f a second. 

We have introduced here a statement of these minute subdivisions, 
because they form a natural counterpart to the immense periods which we 
shall soon have to consider, and are, with the latter, curiously illustrative 
of a fundamental trait of Hindu character : a fantastic imaginativeness, 
which delights itself with arbitrary theorizings, and is unrestrained by, 
and careless of, actual realities. Thus, having no instruments by which 
they could measure even seconds with any tolerable precision, they vied 
with one another in dividing the second down to the farthest conceivable 
limit of minuteness ; thus, seeking infinity in the other direction also, 
while they were almost destitute of a chronology or a history, and could 
hardly fix with accuracy the date of any event beyond the memory of 
the living generation, they devised, and put forth, as actual, a frame- 
work of chronology reaching for millions of millions of years back into 
the past and forward into the future. 

12. ... Of thirty of these sidereal days is composed a month ; 
a civil (sdvana) month consists of as many sunrises ; 

18. A lunar month, of as many lunar days {tithi); a solar 
(sdura) month is determined by the entrance of the sun into a 
sign of the zodiac : twelve months make a year. . . . 

We have here described days of three different kinds, and months 
and years of four ; since, according to the commentary, the last clause 
translated means that twelve months of each denomination make up a 
year of the same denomination. Of some of these, the practical use 
and value will be made to appear later ; but as others are not elsewhere 
referred to in this treatise, and as several are merely arbitrary divisions 
of time, of which, so far as we can discover, no use has ever been made, 
it may not be amiss briefly to characterize them here. 

Of the measures of time referred to in the twelfth verse, the day is 
evidently the starting-point and standard. The sidereal day is the time 
of the earth's revolution on its axis ; data for determining its length are 
given below, in v. 34, but it does not enter as an element into the later 
processes. Nor is a sidereal month of thirty sidereal days, or a sidereal 
year of three hundred and sixty such days (being less than the true 
sidereal year by about six and a quarter sidereal days), elsewhere men- 



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i. 13.] Stirya-Siddhdnta. 151 

tioned in this work, or, so far as we know, made account of in any 
Hindu method of reckoning time. The oivil (sdvana) day is the natural 
day : it is counted, in India, from sunrise to sunrise (see below, v. 36), 
and is accordingly of variable length : it is, of course, • an important 
element in all computations of time. A month of thirty, and a year of 
three hundred and sixty, such days, are supposed to have formed the 
basis of the earliest Hindu chronology, an intercalary month being added 
once in five years. This method is long since out of use, however, and 
the month and year referred to here in the text, of thirty and three 
hundred and sixty natural days respectively, without intercalations, are 
elsewhere assumed and made use of only in determining, for astrological 
purposes, the lords of the month and year (see below, v. 52). 

The standard of the lunar measure of time is the lunar month, the 
period of the moon's synodical revolution. It is reckoned either from 
new-moon to new-moon, or from full-moon to full-moon ; generally, tho 
former is called mukhya^ " primary," and the latter g&una, " secondary " : 
but, according to our commentator, either of them may be denominated 
primary, although in fact, in this treatise, only the first of them is so 
regarded ; and the secondary lunar month is that which is reckoned 
from any given lunar day to the next of the same name. This* natural 
month, containing about twenty-nine and a half days, mean solar time, 
is then divided into thirty lunar days (tithi), and this division, although 
of so unnatural and arbitrary a character, the lunar days beginning and 
ending at any moment of the natural day and night, is, to the Hindu, 
of the most prominent practical importance, since by it are regulated 
the performance of many religious ceremonies (see below, xiv. 13), and 
upon it depend the chief considerations of propitious and unpropitious 
times, and the like. Of the lunar year of twelve lunar months, how- 
ever, we know of no use made in India, either formerly or now, except 
as it has been introduced and employed by the Mohammedans. 

Finally, the year last mentioned, the solar year, is that by which time 
is ordinarily reckoned in India. It is, however, not the tropical solar 
year, which we employ, but the sidereal, no account being made of the 
precession of the equinoxes. The solar month is measured by the con- 
tinuance of the sun in each successive sign, and varies, according to the 
rapidity of his motion, from about twenty-nine and a third, to a little 
more than thirty-one and a half, days. There is no day corresponding 
to this measure of the month and of the year. 

In the ordinary reckoning of time, these elements are variously com- 
bined. Throughout Southern India (see Warren's Kala Sankalita, 
Madras: 1825, p. 4, etc.), the year and month made use of are the 
solar, and the day the civil ; the beginning of each month and year 
being counted, in practice, from the sunrise nearest to the moment of 
their actual commencement. In all Northern India the year is luni- 
solar ; the month is lunar, and is divided into both lunar and civil days ; 
the year is composed of a variable number of months, either twelve or 
thirteen, beginning always with , the lunar month of which the com- 
mencement next precedes the true commencement of the sidereal year. 
But, underneath this division, the division of the actual sidereal year 
into twelve solar months is likewise kept up, and to maintain the con- 



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152 JK Burgess, etc., p. 13- 

currence of the civil and lunar days, and the lunar and solar months, is 
a process of great complexity, into the details of which we need not 
enter here (see Warren, as above, p. 57, etc.). It will be seen later in 
this chapter (w. 48-61) that the Sftrya-Siddh&nta reckons time by this 
latter system, by the combination of civil, lunar, and sidereal elements. 

13. . . . This is called a day of the gods. 

14. The day and night of the gods and of the demons are 
mutually opposed to one another. Six times sixty of them are 
a year of the gods, and likewise of the demons. 

" This is called," etc. : that is, as the commentary explains, the year 
composed of twelve solar months, as being those fast mentioned ; the 
sidereal year. It appears to us very questionable whether, in the first 
instance, anything more was meant by calling the year a day of the 
gods than to intimate that those beings of a higher order reckoned time 
upon a grander scale : just as the month was said to be a day of the 
Fathers, or Manes (xiv. 14), the Patriarchate (v. 18), a day of the 
Patriarchs (xiv. 21), and the JEjou (v. 20), a day of Brahma; all these 
being familiar Puranic designations. In the astronomical reconstruction 
of the Puranic system, however, a physical meaning has been given to 
this day of the gods : the gods are made to reside at the north pole, and 
the demons at the south ; and then, of course, during the half-year 
when the sun is north of the equator, it is day to the gods and night to 
the demons ; and during the other half-year, the contrary. The subject 
is dwelt upon at some length in the twelfth chapter (xii. 45, etc.). 
To make such a division accurate, the year ought to be the tropical, and 
not the sidereal ; but the author of the Surya-Siddh&nta has not yet 
begun to take into account the precession. See what is said upon this 
subject in the third chapter (vv. 9-10). 

The year of the gods, or the divine year, is employed only in des- 
cribing the immense periods of which the statement now follows. 

15. Twelve thousand of these divine years are denominated 
a Quadruple Age {caiuryuga) ; of ten thousand times four hun- 
dred and thirty-two solar years 

16. Is composed that Quadruple Age, with its dawn and twi- 
light. The difference of the Golden and the other Ages, as 
measured by the difference in the number of the feet of Virtue 
in each, is as follows : 

17. The tenth part of an Age, multiplied successively by four, 
three, two, and one, gives the length of the Golden and the other 
Ages, in order : the sixth part of each belongs to its dawn and 
twilight. 

The period of 4,320,000 years is ordinarily styled Great Age (ma- 
K&yuga), or, as above in two instances, Quadruple Age (caturyuga). 
In the S&rya-Siddh&nta, however, the former term is not once found, 
and the latter occurs only in these verses ; elsewhere, Age (yuga) alone 
is employed to denote it ; and always denotes it, unless expressly limited 
by the name of the Golden (krta) Age. 



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L 11.] Mrya-Siddh&nta. 158 

The composition of the Age, or Great Age, is then as follows : 

Divine yean. Solar yean. 

Dawn, 4<x> i44,ooo 

Golden Age (hrta yuga) t 4ooo 1,440,000 

Twilight, 4oo 144,000 

Total duration of the Golden Age, 4,800 1,728,000 

Dawn, 3oo 108,000 

Silver Age (tretd yvga), 3ooo 1,080,000 

Twilight, 3oo x 08,000 

Total duration of the Silver Age, 3,6oo 1,296,000 

Dawn, 200 72,000 

Brazen Age (dvdpara yuga), 2000 720,000 

Twilight, 200 72,000 

Total duration of the Brazen Age, 2,400 864,000 

Dawn, 100 36,ooo 

Iron Age (kali yuga\ 1000 36o,ooo 

Twilight, 100 36,ooo 

Total duration of the Iron Age, 1,200 43a,ooo 

Total duration of a Great Age, 12,000 4,3ao,ooo 

Neither of the names of the last three ages is once mentioned in the 
Surya-Siddhanta. The first and last of the fonr are derived from the 
game of dice : krta, " made, won," is the side of the die marked with 
four dots — the lucky, or winning one ; kali is the side marked with one 
dot only — the unfortunate, the losing one. In the other names, of 
which we do not know the original and proper meaning, the numerals 
tri, u three," and dv&, " two," are plainly recognizable. The relation 
of the numbers four, three, two, and one, to the length of the several 
periods, as expressed in divine years, and also as compared with one 
another, is not less clearly apparent The character attached to the 
different Ages by the Hindu mythological and legendary history so 
closely resembles that which is attributed to the Golden, Silver, Brazen, 
and Iron Ages, that we have not hesitated to transfer to them the latter 
appellations. An account of this character is given in Manu i. 81-86. 
During the Golden Age, Virtue stands firm upon four feet, truth and 
justice abound, and the life of man is four centuries ; in each following 
Age Virtue loses a foot, and the length of life is reduced by a century, 
so that in the present, the Iron Age, she has but one left to hobble 
upon, while the extreme age attained by mortals is but a hundred years. 
See also Wilson's Vishnu rurana, p. 622, eta, for a description of the 
vices of the Iron Age. 

This system of periods is not of astronomical origin, although the 
fixing of the commencement of the Iron Age, the only possibly his- 
torical point in it, is, as we shall see hereafter, the result of astro- 
nomical computation. Its arbitrary and artificial character is apparent. 
It is the system of the Puranas and of Manu, a part of the received 
Hindu cosmogony, to which astronomy was compelled to adapt itself. 
vol. vi. 20 



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154 E^Burgess, etc., [i. 17- 

We ought to remark, however, that in the text itself of Manu (i. 68-71) 
the duration of the Great Age, called by him Divine Age, is given as 
twelve thousand years simply, and that it is his commentator who, by 
asserting these to be divine years, brings Manu's cosmogony to an agree- 
ment with that of the Puranas. This is a strong indication that the 
divine year is an afterthought, and that the period of 4,320,000 years 
is an expansion of an earlier one of 12,000. Vast as this period is, 
however, it is far from satisfying the Hindu craving after infinity. We 
are next called upon to construct a new period by multiplying it by a 
thousand. 

18. One and seventy Ages are styled here a Patriarchate 
{rnanvaniura) ; at its end is said to be a twilight which has the 
number of years of a Golden Age, and which is a deluge, 

19. In an jEon (kalpa) are reckoned fourteen such Patriarchs 
(manu) with their respective twilights ; at the commencement of 
the jEon is a fifteenth dawn, having the length of a Golden 
Age. 

The -£Con is accordingly thus composed : 

Divine yean. Solar years. 

The introductory dawn, - 4,8oo 1,728,000 

Seventy-one Great Ages, • 85 2,000 306,720,000 
A twilight, 4,800 1,728,000 

Duration of one Patriarchate, 856,8oo 3o8,448,ooo 

Fourteen Patriarchates, 11,995,200 4,318,272,000 

Total duration of an JE<m t 12,000,000 4,320,000,000 

Why the factors fourteen and seventy-one were thus used in making 
tip the JSon is not ohvious ; unless, indeed, in the division by fourteen 
it to be recognized the influence of the number seven, while at the 
same time such a division furnished the equal twilights, or interme- 
diate periods of transition, which the Hindu theory demanded. The 
system, however, is still that of the Pur&nas (see Wilson's Yish. Pur. p. 
24, etc.) ; and Manu (i. 72, 79) presents virtually the same, although he 
has not the term JSon (kalpa), but states simply that a thousand Divine 
Ages make up a day of Brahma, and seventy-one a Patriarchate. The 
term manvantara, " patriarchate," means literally " another Manu," or, 
" the interval of a Manu." Manu, a word identical in origin and mean- 
ing with our " man," became to the Hindus the name of a being per- 
sonified as son of the Sun (Vivasvant) and progenitor of the human 
race. In each Patriarchate there arises a new Manu, who becomes for 
his own period the progenitor of mankind (see Wilson's Vish. Pur. p. 
24). 

20. The JEon, thus composed of a thousand Ages, and which 
brings about the destruction of all that exists, is styled a day of 
Brahma; his night is of the same length. 

21. His extreme age is a hundred, according to this valuation 
of a day and a night. . . . 



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i. 23.] S&rya-Siddli&nta. 155 

We have already found indications of an assumed destruction of 
existing things at the termination of the lesser periods called the Age 
and the Patriarchate, in the necessity of a new revelation of virtue and 
knowledge for every Age, and of a new father of the human race for 
every Patriarchate. These are left, it should seem, to show us how the 
systcnf of cosmical periods grew to larger and larger dimensions. The 
full development of it, as exhibited in the Puranas and here, admits only 
two kinds of destruction : the one occurring at the end of each ^Eon, 
or day of Brahma, when all creatures, although not the substance of the 
world, undergo dissolution, and remain buried in chaos during his night, 
to be created anew when his day begins again ; the other taking place 
at the end of Brahma's life, when all matter even is resolved into its 
ultimate source. 

According to the commentary, the "hundred" in verse 21 means a hun- 
dred, years, each composed of three hundred and sixty days and nights, 
and not a hundred days and nights only, as the text might be understood 
to signify ; since, in all statements respecting age, years are necessarily 
understood to be intended. The length of Brahma's life would be, 
then, 864,000,000,000 divine years, or 311,040,000,000,000 solar years. 
This period is also called in the Puranas a para, " extreme period, and 
ite half &})ar&rdha (see Wilson's Vish. Pur. p. 25) ; although the latter 
term has obtained also an independent use, as signifying a period still 
more enormous (ibid. p. 630). It is curious that the commentator does 
not seem to recognize the affinity with this period of the expression 
used in the text, param dyuh, " extreme age," but gives two different 
explanations of it, both of which are forced and unnatural. 

The author of the work before us is modestly content with the number 
of years thus placed at his disposal, and attempts nothing farther. So 
is it also with the Puranas in general ; although some of them, as the 
Vishnu (Wilson, p. 637) assert that two of the greater pardrdhas con- 
stitute only a day of Vishnu, and others (ibid. p. 25) that Brahma's 
whole life is but a twinkling of the eye of Krshna or of Qiva. 

21 The half of his life is past ; of the remainder, this is 

the first JSon. 

22. And of this JSon, six Patriarchs (manu) are past* with 
their respective twilights; and of the Patriarch Manu son of 
Vivasvant, twenty-seven Ages are past; 

23. .Of the present, the twenty-eighth, Age, this Golden Age 
is past : from this point, reckoning up the time, one should com- 
pute together the whole number. 

The designation of the part already elapsed of this immense period 
seems to be altogether arbitrary. It agrees in general with that given 
in the Puranas, and, so far as the Patriarchs and their periods are con* 
cemed, with Manu also. The name of the present JEon is V&r&ha, 
" that of the boar," because Brahma, in performing anew at its com- 
mencement the act of creation, put on the form of that animal (see 
Wilson's Vish. Pur. p. 2*7, etc.). Tne one preceding is called the PMma, 
"that of the lotus. This nomenclature, however, is not universally 



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166 JB. Burgess, etc., [i. 28- 

accepted : under the word kalpa, in the Lexicon of Bobtlingk and Roth, 
may be found another system of names for these periods. Mann (i. 61, 
62) gives the names of the Patriarchs of the past Patriarchates ; the 
Puranas add other particulars respecting them, and also respecting those 
which are still to come (see Wilson's Vish. Pur. p. 259, etc.). 

The end of the Golden Age of the current Great Age is the lime at 
which the Surya-Siddhanta claims to have been revealed, and the epoch 
from which its calculations profess to commence. We will, accordingly, 
as the Sun directs, compute the number of years which are supposed to 
have elapsed before that period. 

Drrtae yean. Bohr year*. 

Dawn of current JEon, 48oo 1,720,000 

Six Patriarchates, 5,i4o,8oo i,85o,688,ooo 

Twenty-seven Great Ages, 3a4>ooo ii6,§4o>ooo 

Total till commencement of present Great Age, 6,469,600 1 ,969,056,000 

Golden Age of present Great Age, 4,8oo 1,728,60b 

Total time elapsed of current J5on, 5,474,4oo 1,970,784,000 

Half Brahma's life, 432,000,000,000 i55,52o,ooo,ooo,ooo 

As the existing creation dates from the commencement of the current 
JSon, the second of the above totals is the only one with which the 
Surya-Siddhanta henceforth has any thing to do. 

We are next informed that the present order of things virtually began 
at a period less distant than the commencement of the jEon. 

24. One hundred times four hundred and seventy-four divine 
years passed while the All- wise was employed in creating the 
animate and inanimate creation, plants, stars, gods, demons, and 
the rest. * 

That is to say : 

Divine years. 8olar years. 

From the total above given, 5,474i4oo 1,970,784,000 

deduct the time occupied in creation, 47«4oo 17,064,000 

the remainder is 5,4?7,ooo 1,953,720,000 

This, then, is the time elapsed from the true commencement of the ex- 
isting order of things to the epoch of this work. The deduction of this 
period as spent by the Deity in the work of creation is a peculiar feature 
of the Surya-Siddhanta. We shall revert to it later (see below, under 
vv. 29-34), as its significance cannot be shown 'until other data are 
before us. 

25. The planets, moving westward with exoeeding velocity, 
but constantly beaten by the asterisms, fall behind, at a rate 
precisely equal, proceeding each in its own path. 

26. Hence they have an eastward motion. From the number 
of their revolutions is derived their daily motion, which is dif- 
ferent according to the size of their orbits; in proportion to this 
daily motion they pass through the asterisms. 



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L 27.] S&ryorSiddkdnta. 157 

. 27. One which moves swiftly passes through them in a short 
time ; one which moves slowly, m a long time. By their move- 
ment, the revolution is accounted complete at the end of the 
asterism Kevatf . 

We have here presented a patt of the physical theory of the planetary 
motions, that which accounts for the mean motions : the theory is sup- 
plemented by the explanation given in the next chapter of the disturbing 
forces which give rise to the irregularities of movement The earth is a 
sphere, and sustained immovable in the centre of the universe (xii. 82), 
while all the heavenly bodies, impelled by winds, or vortices, called pro- 
vectors (ii. 3), revolve about it from east to west. In this general west- 
ward movement, the planets, as the commentary explains it, are, owing 
to their weight and the weakness of their vortices, beaten by the aster- 
isms (nalxhatra or bha> the groups of atari constituting the lunar man- 
sions {see below, chapter viii], and used here, as in various other places, 
to designate the whole firmament of fixed stars), and accordingly fall 
behind (lambante—labuntur, delabuntur), as if from shame: and this is 
the explanation of their eastward motion, which is only apparent and rela- 
tive, although wont to be regarded as real by those who do not under- 
stand the true causes of things. But now a new element is introduced 
in|p the theory, which does not seem entirely consistent with this view of 
the merely relative character of the eastward motion. It is asserted that 
the planets lag behind equally, or that each, moving in its own orbit, 
loses an equal amount daily, as compared with the asterisms. And we 
shall find farther on (xii. 78-89) that the dimensions of the planetary 
orbits are constructed upon this sole principle, of making the mean daily 
motion of each planet eastward to be the same in amount, namely 
11,858.717 yqjanas: the amount of westward motion being equal, in 
eAch case, to the difference between this amount and the whole orbit of 
the planet Now if the Hindu idea of the symmetry and harmony of the 
universe demanded that the movements of the planets should be equal, it 
was certainly a very awkward and unsatisfactory way of complying with 
that demand to make the relative motions alone, as compared with the 
fixed stars, equal, and the real motions so vastly different from one an- 
other. We should rather expect that some method would have been de- 
vised for making the latter come out alike, and the former unlike, and the 
result of differences in the weights of the planets and the forces of the 
impelling currents. It looks as if this principle, and the conformity to it 
of the dimensions of the orbits, might have come from those who regarded 
the apparent daily motion as the real motion. But we know that Arya- 
bhatta held the opinion that the earth revolved upon its axis, causing 
thereby the apparent westward motion of the heavenly bodies (see Cole- 
brooke's Hindu Algebra, p. xxxviii ; Essays, ii. 467), and so, of course, 
that the planets really moved eastward at an equal rate among the stars ; 
and although the later astronomers are nearly unanimous against him, 
we cannot help surmising that the theory of the planetary orbits ema- 
nated from him or his school, or from some other of like opinion* It is 
not upon record, so far as we are aware, that any Hindu astronomer, of 
any period, held, as did some of thejftreek philosophers (see Whewell'a 
History of the Inductive Sciences, B. V. ch. i), a heliocentric theory. 



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158 27. Burgess, etc., [L 27- 

The absolute motion eastward of all the planets being equal, their 
apparent motion is, of coarse, in the (inverse) ratio of their distance, or 
of the dimensions of their orbits. 

The word translated u revolution " is bliagana, literally " troop of aster- 
isms ;" the verbal root translated " pass through " is bhuj, " enjoy," from 
which comes also the common term for the daily motion of a planet, 
bhukti, literally ** enjoyment" When a planet has M enjoyed the whole 
troop of asterisms," it has made a complete revolution. 

Tne initial point of the fixed Hindu sphere, from which longitudes are 
reckoned, and at which the planetary motions are held by all the schools 
of Hindu astronomy to have commenced at the creation, is the end of 
the asterism Revatl, or the beginning of Ac, vinl (see chapter viii. for a 
full account of the asterisms). Its situation is most nearly marked by 
that of the principal star of Revatl, which, according to the Snrya- 
Siddhanta, is 10' to the west of it, but according to other authorities 
exactly coincides with it That star is by all authorities identified with 
{ Piscium, of which the longitude at present, as reckoned by us, from the 
vernal equinox, is 17° 54'. Making due allowance for the precession, we 
find that it coincided in position with the vernal equinox not far from the 
middle of the sixth century, or about A. D. 570. As such coincidence 
was the occasion of the point being fixed upon as the beginning of the 
sphere, the time of its occurrence marks approximately the era of /he 
fixation of the sphere, and of the commencement of the history of modern 
Hindu astronomy. We say approximately only, because, in the first 
place, as will be shown in connection with the eighth chapter, the accu- 
racy of the Hindu observations is not to be relied upon within a degree ; 
and, in the second place, the limits of the asterisms being already long 
before fixed, it was necessary to take the beginning of some one of them 
as that of the sphere, and the Hindus may have regarded that of Acvinl 
as sufficiently near to the equinox for their purpose, when it was, in fact, 
two or three degrees, or yet more, remote from it, on either side ; and 
each degree of removal would correspond to a difference in time of about 
seventy years. 

In the most ancient recorded lists of the Hindu asterisms (in the texts 
of the Black Yajur-Veda and of the Atharva-Veda), Krttika, now the 
third, appears as the first The time when the beginning of that aster- 
ism coincided with the vernal equinox would be nearly two thousand 
years earlier than that given above for the coincidence with it of the first 
point of Acvinl. 

28. Sixty seconds (vikald) make a minute (kald) ; sixty of 
these, a degree {bhdga) ; of thirty of the latter is composed a 
sign (rdfi) ; twelve of these are a revolution (bhagana). 

The Hindu divisions of the circle are thus seen to be the same with 
the Greek and with our own, and we shall accordingly make use, in 
translating, of our own familiar terms. Of the second (vikald) very little 
practical use is made ; it is not more than two or three times alluded to 
in all the rest of the treatise. The minute (kald) is much more often 
called lipid (or liptikd) ; this is not a n original Sanskrit word, but was 
borrowed from the Greek max ov.WThe degree is called either bhdga or 
anpz; both words, like the equivalent Greek word /i«pa, mean a u part, 



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i. 34.] 8&rya-SiddJidnta. 159 

portion," The proper signification of rdfi, translated " sign," is simply 
" heap, quantity ;" it is doubtless applied to designate a sign as being a 
certain number, or sum, of degrees, analogous to the use of gana in 
bhagana (explained above, in the last note), and of rdpi itself in dinar&fi, 
"sum of days" (below, v. 53). In the Hindu description of an arc, the 
sign is as essential an element as the degree, and no arcs of greater length 
than thirty degrees are reckoned in degrees alone, as we are accustomed 
to reckon them. The Greek usage was the same. We shall hereafter 
see that the signs into which any circle of revolution is divided are named 
Aries, Taurus, etc., beginning from the point which is regarded as the 
starting point ; so that these names are applied simply to indicate the 
order of succession of the arcs of thirty degrees. 

29. In an Age (yuga), the revolutions of the sun, Mercury, 
and Venus, and of the conjunctions (<fyhra) of Mars, Saturn, 
and Jupiter, moving eastward, are four million, three hundred 
and twenty thousand; 

30. Of the moon, fifty -seven million, seven hundred and fifty- 
three thousand, three hundred and thirty-six; of Mars, two 
million, two hundred and ninety-six thousand, eight hundred 
and thirty-two ; 

31. Of Mercury's conjunction (tfghraX seventeen million, nine 
hundred and thirty-seven thousand, ana sixty ; of Jupiter, three 
hundred and sixty-four thousand, two hundred and twenty ; 

32. Of Venus's conjunction {pghra), seven million, twenty-two 
thousand, three hundred and seventy-six ; of Saturn, one hun- 
dred and forty -six thousand, five hundred and sixty-eight; 

83. Of the moon's apsis (ucca), in an Age, four hundred and 
eighty-eight thousand, two hundred and three ; of its node (pdta\ 
in the contrary direction, two hundred and thirty-two thousand, 
two hundred and thirty-eight ; 

34. Of the asterisms, one billion, five hundred and eighty-two 
million, two hundred and thirty-seven thousand, eight hundred 
and twenty-eight 

These are the fundamental and most important elements upon which 
is founded the astronomical system of the Surya-Siddhanta, We present 
them below in a tabular form, but must firat explain the character of 
some of them, especially of some of those contained in verse 29, which 
we have omitted from the table. 

The revolutions of the sun, and of Mars, Jupiter, and Saturn, require 
no remark, save the obvious one that those of the sun are in fact sidereal 
revolutions of the earth about the sun. To the sidereal revolutions of 
the moon we add also her synodical revolutions, anticipated from the next 
following passage (see v. 35). By the moon's "apsis" is to be under- 
stood her apogee ; ucca is literally " height," i. e. w extreme distance :" 
the commentary explains it by mandocca, " apex of slowest motion :" as 
the same word is used to designate the aphelia of the planets, we were 
obliged to take in translating it the indifferent term apsis, which applies 
equally to both geocentric and heliocentric motion. The "node" is the 
ascending node (see ii. 7) ; the dual "nodes" is never employed in this 



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160 JE Burgess, etc., [i. 34. 

work. But the apparent motions of the planets are greatly complicated 
by the fact, unknown to the Greek and the Hindu, that they are revolv- 
ing about a centre about which the earth also is revolving. When any 
planet is on the opposite side of the sun from us, and is accordingly mov- 
ing in space in a direction contrary to ours, the effect of our change of 
place is to increase the rate of its apparent change of place ; again, when 
it is upon our side of the sun, and moving in the same direction with us, 
the effect of our motion is to retard its apparent motion, and even to 
cause it to seem to retrograde. This explains the " revolutions of the 
conjunction" of the three superior planets : their " conjunctions" revolve 
at the same rate with the earth, being always upon the opposite side of 
the sun from us; and when, by the combination of its own proper 
motion with that of its conjunction, the planet gets into the latter, its 
rate of apparent motion is greatest, becoming less in proportion as it 
removes from that position. The meaning of the word which we have 
translated "conjunction" is "swift, rapid:" a literal rendering of it 
would be "swift-point," or "apex of swiftest motion;" but, after 
much deliberation, and persevering trial of more than one term, we have 
concluded that "conjunction" was the least exceptionable word by 
which we could express it. In the case of the inferior planets, the 
revolution of the conjunction takes the place of the proper motion of 
the planet itself! By the definition given in verse 27, a planet must, in 
order to complete a revolution, pass through the whole zodiac; this 
Mercury and Venus are only able to do as they accompany the sun in 
his apparent annual revolution about the earth. To the Hindus, too, 
who had no idea of their proper movement about the sun, the annual 
motion must have seemed the principal one; and that by virtue of which, 
in their progress through the zodiac, they moved now faster and now 
slower, must have appeared only of secondary importance. The term 
" conjunction," as used in reference to these planets, must be restricted, 
of course, to the superior conjunction. The physical theories by which 
the effect of the conjunction (ptgfira) is explained, are given in the next 
chapter. In the table that followB we have placed opposite each planet 
its own proper revolutions only. 

It is farther to be observed that all the numbers of revolutions, ex- 
cepting those of the moon's apsis and node, are divisible by four, so 
that, properly speaking, a quarter of an Age, or 1,080,000 years, rather 
than a wnole Age, is their common period. This is a point of so muoh 
importance in the system of the Surya-Siddh&nta, that we have added, 
in a second column, the number of revolutions in the lesser period. 

In the third column, we add the period of revolution of each planet, 
as found by dividing by the number of revolutions of each the number 
of civil days in an Age (which is equal to the number of sidereal days, 
given in v. 34, diminished by the number of revolutions of the sun ; see 
below, v. 37) ; they are expressed in days, n&dis, vin&dls, and respira- 
tions ; the latter may be converted into sexagesimals of the third order 
by moving the decimal point one place farther to the right 

In the fourth column are given the mean daily motions. 

We shall present later some comparison of these elements with those 
adopted in other systems of astronomy, ancient and modern. 



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i 34.] 



S&rya-Siddhdnta. 



161 



Mean Motions of the Planet*. 



Planet 


Number of 
revolatioM in 


Namber of 
revolution* in 


Length of a revolution 
in mean solar tame. 


Mean daily motion. 




4,330,000 yoan. 


1,090,000 year* 




Son, 


4320,000 


1,080,000 


d n * p 
365 i5 3i 3.i4 


• e ti in tut 

59 8 10 10.4 


nonary, 


17,937,060 


4,484265 


87 58 10 5.57 


4 5 32 20 4i-9 


Venus, 


7,022,376 


1,755,594 


224 4i 54 5.o6 


1 36 7 43 37.3 


Man, 


2,996,832 


574208 


686 59 5o 5.87 


3i 26 28 1 1.1 


Jupiter, 


364,220 


9i,o55 


4,332 19 14 2.09 


4 59 8 48.6 


Saturn, 


i46,568 


36,642 


10,765 46 23 0.41 


2 22 53.4 


Moon: 










eider, rev. 


57,753,336 


14,438,334 


27 19 18 0.16 


i3 10 34 52 3.8 


tynod. rev. 


53,433,336 


13,358,334 


29 3i 5o 0.70 


12 11 26 4i 534 


rev. of apsis, 


488,2o3 


I22,o5o$ 


3,232 5 37 1.36 


6 4o 58 42.5 


" u node, 


232,238 


58,o5o} 


6,794 23 59 2.35 


3 10 44 43.2 



The arbitrary and artificial method in which the fundamental ele- 
ments of the solar system are here presented is not peculiar to the 
Surya-&iddhanta; it is also adopted by all the other text-books, and is 
to be regarded as a characteristic feature of the general astronomical 
system of the Hindus. Instead of deducing the rate of motion of each 
planet from at least two recorded observations of its place, and estab- 
lishing a genuine epoch, with the ascertained position of each at that 
time, they start with the assumption that, at the beginning of the 
present order of things, all the planets, with their apsides and nodes, 
commenced their movement together at that point in the heavens (near 
? Fiacram, as explained above, under verse 27) fixed upon as the initial 
point of the sidereal sphere, and that they return, at certain fixed inter- 
vals, to a universal conjunction at the same point As regards, however, 
the time when the motion commenced, the frequency of recurrence 
of the conjunction, and the date of that which last took place, there is 
discordance among the different authorities. With the Surya>Sid- 
• dhanta, and the other treatises which adopt the same general method, 
the determining point of the whole system is the commencement of the 
current Iron Age (kali yugd) ; at that epoch the planets are assumed to 
have been in mean conjunction for the last time at the initial point of 
the sphere, the former conjunctions having taken place at intervals of 
1,080,000 years previous. The instant at which the Age is made to 
commence is midnight on the meridian of Ujjayini (see below, under v. 
62), at the end of the 588,465th and beginning of the 588,466th day 
(civil reckoning) of the Julian Period, or between the 17th and 18th of 
February 1612 J.P., or 3102 £. C. (see below, under vv. 45-53, for the 
computation of the number of days since elapsed). Now, although no 
voch conjunction as that assumed by the Hindu astronomers ever did 
or ever will take place, the planets were actually, at the time stated, 
approximating somewhat nearly to a general conjunction in the neigh- 
borhood of the initial point of the Hindu sphere ; this is shown by the 
next table, in which we give their actual mean positions with reference 
to that point (including also those of the moon's *pojgee *°d node) ; 
they have been obligingly furnished us by Prof. Winlock, Superi** 



VOL. VI. 



21 



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162 



K Burgess, etc., 



[t. 34. 



. tendent of the American Ephemeris and Nautical Almanac. The posi- 
tions of the primary planets are obtained by LeVerrier's times of side- 
real revolution, given in the Annales de rObservatoire, torn, ii (also in 
Biot's Astronomie, 3 me edition, torn, v, 1857), that of the moon by 
Peirce's tables, and those of its apogee and node by Hansen's Tables de 
la Lune. The origin of the Hindu sphere is regarded as being 18° 5' 
8" east of the vernal equinox of Jan. 1, I860, and 50° 22' 29' r west of 
that of Feb. 17, 3102 B. C, the precession in the interval being 68° 27' 
37". We add, in a second column, the mean longitudes, as reckoned 
from the vernal equinox of the given date, for the sake of comparison 
with the similar data given by Bentley (Hind. Ask, p. 125) and by 
Bailly(Ast Ind. et Or., pp. Ill, 182), which we also subjoin. 

Positions of the Planets, midnight, at Ujjayint, Feb. 17-18, 3102 B. C. 



Planet 


From beginning 
of Hindu sphere. 


Longitude. 


Bentley. 


B&UJy. 


Sun, 


• • »# 
- 7 5i 48 


/ u 

3oi 45 43 


• « »» 
3oi 1 1 


• » 11 
3oi 5 57 


Mercury, 


-4i 3 o6 


268 34 5 


267 35 26 


261 i4 ai 


Venus, 


+ 24 58 59 


334 36 3o 


333 44 3 7 


334 aa 18 


Mars, 


- 19 49 26 


289 48 5 


288 55 19 


388 55 56 


Jupiter, 


+ 8 38 36 


3i8 16 ^ 7 


3i8 3 54 


3io aa 10 


Saturn, 


-28 1 i3 


281 36 '18 


280 1 58 


293 8 21 


Moon, 


- 1 33 4i 


3o8 3 5o 


3o6 53 42 


3oo 5i 16 


do. apsis, 


+ 95 19 21 


44 56 42 


61 12 26 


61 i3 33 


do. node, 


+198 a4 45 


148 2 16 


i44 38 32 


i44 37 4i 



The want of agreement between the results of the three different in- 
vestigations illustrates the difficulty and uncertainty even yet attending 
inquiries into the positions of the heavenly bodies at so remote an 
epoch. It is very possible that the calculations of the astronomers who 
were the framers of the Hindu system may have led them to suppose 
the Approach to a conjunction nearer than it actually was ; bat, however 
that may be, it seems hardly to admit of a doubt that the epoch was 
arrived, at by astronomical calculation carried backward, and that it was 
fixed upon as the date of the last general conjunction, and made to 
determine the commencement of the present Age of the world, because 
the errors of the assumed positions of the planets at that time would 
be so small, and the number of years since elapsed so great, as to make 
the errors in the mean motions into which those positions entered as 
an element only trifling in amount 

The moon's apsis and node, however, were treated in a different 
manner. Their distance from the initial point of the sphere, as shown 
by the table, was too great to be disregarded. They were accordingly 
ingly exempted from the general law of a conjunction once in 1,080,000 
years, and such a number of revolutions was assigned to them as should 
make their positions at the epoch come out, the one a quadrant, the 
other a half-revolution, in advance of the initial point of the sphere. 

We can now see why the deduction spoken of above (v. 24), for time 
spent in creation, needed to be made. In order to bring all the planet* 
to a position #f mean conjunction at the epoch, the time previously 



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i. 34.] Surya-Siddh&nta. 163 

elapsed must be an exact multiple of the lesser period of 1,080,000 years, 
or the quarter- Age ; in order to give its proper position to the moon's 
apsis, that time must contain a certain number of whole Ages, which 
are the periods of conjunction of the latter with the planets, together 
with a remainder of three quarter- Ages ; for the moon's node, in like 
manner, it must contain a certain number of half-Ages, with a remainder 
of one quarter- Age. Now the whole number of years elapsed between the 
beginning of the JEon and that of the current Iron Age is equal to 1826 
-quarter-Ages, with an odd surplus of 864,000 years : from it subtract 
an amount of time which shall contain this surplus, together with three, 
seven, eleven, fifteen, or the like (any number exceeding by three a mul- 
tiple of four), quarter-Ages, and the remainder will fulfil the conditions 
of the problem. The deduction actually made is of fifteen periods + 
the surplus. 

This deduction is a clear indication that, as remarked above (under 
v. 17), the astronomical system was compelled to adapt itself to an 
already established Puranic chronology. It could, indeed, fix the pre- 
viously undetermined epoch of the commencement of the Iron Age, but 
it oould not alter the arrangement of the preceding periods* 

It is evident that, with whatever accuracy the mean positions of the 
planets may, at a given time, be ascertained by observation by the 
Hindu astronomers, their false assumption of a conjunction at the epoch 
of 8102 B. C. must introduce an element of error into their determina- 
tion of the planetary motions. The annual amount of that error may 
indeed be small, owing to the remoteness of the epoch, and the great 
number of years among which the errors of assumed position are divi- 
ded, yet it must in time grow to an amount not to be ignored or neglect- 
ed even by observers so inaccurate, and theorists so unscrupulous, as the 
Hindus. This is actually the case with the elements of the Surya-Sid- 
dh&nta; the positions of the planets, as calculated by them for the 
present time, are in some cases nearly 9° from the true places. The 
later astronomers of India, however, have known how to deal with such 
difficulties without abrogating their ancient text-books. As the Surya- 
Biddh&nta is at present employed in astronomical calculations, there are 
introduced into its planetary elements certain corrections, called btja 
(more properly vija ; the word means literally " seed" ; we do not know 
how it arrived at its present significations in the mathematical language). 
That this was so, was known to Davis (As. Res., ii. 236), but he was 
unable to state the amount of the corrections, excepting in the case of 
the moon's apsis and node (ibid. r p. 275). Bentley (Hind. Ask? p. 17&) 
gives them in full; and upon his authority we present them in the 
annexed table. They are in the form, it will be noticed, of additions t©> 
or subtractions from, the number of revolutions given for an Age, and 
the numbers are all divisible by four, in order not to interfere with the 
calculation by the lesser period of 1,080,000 years. Wfe have added 
the corrected number of revolutions, for both the greater and lesser 
period, the corrected time of revolution, expressed in Hindu divisions of 
the day, and the corrected amount of mean daily motion. 

These corrections were first applied, according to Mr. Bentley (As. 
Res^ viii 220), about the beginning of the sixteenth eentury ^ they- are 



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164 



K Burgess^ etc., 



[i. 34. 



Crated by several treatises of that aa well as of later date, not having 
yet superseded by others intended to secure yet greater correctness. 

Mean Motions of the Planets as corrected by the btja. 



1 


Corrected number of revolu- 






1 


Cerree- 


tioiu 


Corrected 


Corrected 


Planet. 


tloo. 


10 4,330,000 
yean. 


iol/XSMMO 
yean. 


time of revolution. 


daily motion. 


Sun, 





4,320,000 


I,o8o,000 


d n ▼ p 
365 i5 3t 3.i4 


59 8 10 1O.4 


Mefoury, 


-16 


17,937,044 


4>484,a6i 


87 58 11 1.26 


4 5 32 19 54.5 


Venus, 


- ia 


7,022,364 


i,755,59i 


224 4i 56 i.35 


1 36 7 43 1.8 


Han, 





2,396,83a 


574,208 


686 59 5o 5.87 


3t 96 38 11. 1 


Jupiter, 


- 8 


354,212 


9i r o53 


, 4,332 24 56 5.56 


4 59 8 24.9 


Saturn, 


+ ia 


i46,58o 


36,645, 
14,438,33^ 


'10,764 53 3o 1. 11 


2 23 28.9 


Moon, 


o 


57,753,336 


27 19 18 0.16 


1 3 10 34 52 3.8 


tt apsis, 


- 4 


488,199 


122,0461 


3,232 7 12 3.37 


6 40 58 30.7 


" node, 


+ 4| a3a,24a 58,o&>* 


6,794 16 58 0.66 


3 10 44 55.0 



We need not, however, rely on' external testimony alone for informa- 
tion as to the period when this correction was made. If the attempt to 
modify the elements in such a manner as to make them givtrThe^true 
positions of the planets at the time when they were so modified was in 
any tolerable degree successful, we ought to be able to discover by cal- 
culation the date of the alteration. If we ascertain for any given time 
the positions of the planets as given by the system, and compare them 
with the true positions as found by our best modern methods, and if we 
then divide the differences of position by the differences in the mean 
motions, we shall discover, in each separate case, when the error was or 
will be reduced to nothing. The results of such a calculation, made for 
Jan. 1, I860, are given below, under v. 67. We see there that, if regard 
is had only to the absolute errors in the positions of the planets, no con- 
clusion of value can be arrived at ; the discrepancies between the dates 
of no error are altogether too great to allow of their being regarded as 
indicating any definite epoch of correction. If, on the other hand, we 
assume the place of the sun to have been the standard by which tho 
positions of the other planets were tested, the dates of no error are seen 
to point quite distinctly to the first half of the sixteenth century as the 
time of the correction, their mean being A. D. 1541. Upon this as- 
sumption, also, we see why no correction of btja was applied to Mars or 
to the moon : the former had, at the given time, only just passed his 
time of complete accordance with the sun, and the motion of the moon 
was also already so closely adjusted to that of the sun, that the differ* 
ence between their errors of position is even now less than 10'. Nor 
is there any other supposition which will explain why the serious error 
in the position of the sun himself was overlooked at the time of the 
general correction, and why, by that correction, the absolute errors of 
position of more than one of the planets are made greater than they 
would otherwise have been, as is the case. It is, in short, clearly evident 
that the alteration of the elements of the Surya-Siddhanta which was 
effected early in the sixteenth century, was an adaptation of the errors 
of position of the other planets to that of the sun, assumed to be cor- 
rect and regarded as the standard. 



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,34.] 



S&rya-Siddkdnta. 



165 



Now if it is possible by this method to arrive approximately at the 
date of a correction applied to the elements of a Biddh&nta, it should 
be possible in like manner to arrive at the date of those elements them- 
selves. For, owing to the false assumption of position at the epoch, 
there is but one point of time at which any of the periods of revolution 
will give the true place of its planet : if, then, as is to be presumed, the 
true places were nearly determined when any treatise was composed, 
and were made to enter as an element into the construction of its 
system, the comparison of the dates of no error will point to the epoch of 
its composition. The method, indeed, as is well known to all those who 
have made any studies in the history of Hindu astronomy, has already 
been applied to this purpose, by Mr. Bentley. It was first originated 
and put forth by him (in vol. vi. of the Asiatic Researches) at a time 
when the false estimate of the age and value of the Hindu astronomy 
presented by Bailly was still the prevailing one in Europe ; he strenu- 
ously defended it against more than one attack (As. Res., viii, and Hind. 
Ask), and finally employed it very extensively in his volume on the 
History of Hindu Astronomy, as a means of determining the age of the 
different Siddh&ntas. We present below the table from which, in the 
latter work (p. 126), he deduces the age of the Surya-8iddh&nta ; the 
column of approximate dates of no error we have ourselves added. 

Bentley** Table of Errors in the Positions of the Planets, as calculated, 
for successive periods, according to the Shrya-SiddMnta. 



PUMt 


Iron Afe 0, 
B.C.3102. 


I. A. 1000, 
B. C. 2102. 


I. A. 2000. 
B. c. uoa 


I. A. 3000, 


I. A. 3639, 


T. A. 4192, 


When 


B.C.102. 


A.D53a 


A. a 1091. 


comet 


Mercury, 


+33 a5 35 


+a5 9 5a 


+16 54 9 


• » M 

+8 38 26 


• 1 •• 
+3 ai 40 


-I ia a8 


A.D. 

945 


Venus, 


-3a 43 36 


-a4 37 3i 


-16 3r 26 


-8 a5 ail-3 i4 45 


+1 14 3 


939 


Mart, 


+12 5 4a 


+ 9 26 3a 


+ 6 47 22 


+4 8 12+a 26 3o 


+0 58 39 


i458 


Jupittr, 


-17 a 53 


-12 44 16 


- 8 a5 39 


-4 7 aj-i a* 47 


+0 4i i4 


906 


Saturn, 


+ao 59 3 


+i5 43 ao 


+10 27 37 


+5 11 54+i 5o 10 


-1 4 aS 


887 


Moon, 


- 5 5a 4i 


- 3 5o48 


- a 9 17 


-0 5a 33'-o 18 3o 


-0 11 


1097 


a apsis, 


-3o 11 a5 


-a3 9 36 


-16* 7 47 


-9 5 58-4 36 a6 


-0 43 10 


no3 


14 node, 


+a3 37 3 1 


+17 59 ai 


+ia 3i 11 +7 3 ij+3 33 19 


+0 3i 5o 


1188 



From an average of the results thus obtained, Bentley draws the con- 
clusion that the Surya-Siddh&nta dates from the latter part of the elev- 
enth century; or, more exactly, A. D. 1091. 

The general soundness of Bentley's method will, we apprehend, be 
denied at the present time by few, and he is certainly entitled to not a 
little credit for his ingenuity in devising it, for the persevering industry 
shown in its application, and for the zeal and boldness with which he 
propounded and defended it He succeeded in throwing not a little 
light upon an obscure and misapprehended subject, and his investiga- 
tions have contributed very essentially to our present understanding of 
the Hindu systems of astronomy. But the details of his work are not 
to be accepted without careful testing, and his general conclusions are 
often unsound, and require essential modification, or are to be rejected 
altogether. This we will attempt to show in connection with his treat- 
ment of the Surya-Siddh&nta. 



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166 K Burgees, etc., p. S4. 

In the first place, Bentley has made a very serious error in that part 
of his calculations which concerns the planet Mercury. As that planet 
was, at the epoch, many degrees behind its assumed place, it was neces- 
sary, of course, to assign to it a slower than its true rate of motion. 
Bat the rate actually given it by the text is not quite enough slower, 
and, instead of exhausting the original error of position in the tenth 
century of our era, as stated by Bentley, would not so dispose of it for 
many hundred years yet to come. Hence the correction of the Mja, as 
reported by Bentley himself, instead of giving to Mercury, as to all the 
rest, a more correct rate of motion, is made to have the contrary effect, 
in order the sooner to run out the original error of assumed position, 
and produce a coincidence between the calculated and the true places of 
the planet 

In the case of the other planets, the times of no error found by 
Bentley agree pretty nearly with those which we have ourselves ob- 
tained, both by calculating backward from the errors of A. D. 1860, 
and by calculating downward from those of B. C. 3102, and which are 
presented in the table given under verse 67. Upon comparing the two 
tables, however, it will be seen at once that Bentley's conclusions are 
drawn, not from the sidereal errors of position of the planets, but from 
the errors of their positions as compared with that of the sun f and that 
of the sun's own error he makes no account at all. This is a method of 
procedure which certainly requires a much fuller explanation and justifi- 
cation than he has seen fit anywhere to give of it The Hindu sphere 
is a sidereal one, and in no wise bound to the movement of the sun. 
The sun, like the other planets, was not in the position assumed for him 
at the epoch of 3102 B. C, and consequently the rate of motion 
assigned to him by the system is palpably different from the real one : 
the sidereal year is about three minutes and a half too long. Why then 
should the sun's error be ignored, and the sidereal motions of the other 
planets considered only with reference to the incorrect rate of motion 
established for him ? It is evident that Bentley ought to have taken 
folly into consideration the sun's position also, and to have shown either 
that it gave a like result with those obtained from the other planets, or, 
if not, what was the reason of the discrepancy. By failing to do so, he 
has, in our opinion, omitted the most fundamental datum of the whole 
calculation, and the one which leads to the most important conclusions. 
We have seen, in treating of the btja, that it has been the aim of the 
modern Hindu astronomers, leaving the sun's error untouched, to amend 
those of the other planets to an accordance with it Now, as things 
are wont to be managed in the Hindu literature, it would be no matter 
for surprise if such corrections were incorporated into the text itself: 
had not the Surya-Siddhanta been, at the beginning of the sixteenth 
century, so widely distributed, and its data so universally known, and 
had not the Hindu science outlived already that growing and productive 
period of its history when a school of astronomy might put forth a cor- 
rected text of an ancient authority, and expect to see it make its way 
to general acceptance, crowding out, and finally causing to disappear, 
the older version — such a process of alteration might, in our view, have 
passed upon it, and such a text might have been handed down to our 



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L84.] S&rychSiddhdnia. 167 

time asBentley would have pronounced, upon internal evidence, to have 
been composed early in the sixteenth century ; while, nevertheless, the 
original error of the sun would remain, untouched and increasing, to in- 
dicate what was the true state of the case. 

But what is the actual position of things with regard to our Bid* 
dh&nta? We find that it presents us a set of planetary elements, which, 
when tested by the errors of position, in the manner already explained, 
do not appear to have been constructed so as to give the true sidereal 
positions at any assignable epoch, but which, on the other hand, exhibit 
evidences of an attempt to bring the places of the other planets into an 
accordance with that of the sun, made sometime in the tenth or eleventh 
century — the precise time is very doubtful, the discrepancies of the 
times of no error being far too great to give a certain result. Now it is 
as certain as anything in the history of Sanskrit literature can be, that 
there was a Surya-Siddh&nta in existence long before that date ; there 
is also evidence in the references and citations of other astronomical 
works (see Colebrooke, Essays, ii. 464 ; Hind. Alg., p. 1) that there have 
been more versions than one of a treatise bearing the tide ; and we have 
seen above, in verse 9, a not very obscure intimation that the present 
work does not present precisely the same elements which had been ac- 
cepted formerly as those of the Surya-Siddhanta* What can lie nearer, 
then, than to suppose that in the tenth or eleventh century a correction 
of btja was calculated for application to the elements of the Siddhanta, 
and was then incorporated into the text, by the easy alteration of four 
or five of its verses ; and accordingly, that while the comparative errors 
of the other planets betray the date of the correction, the absolute error 
of the sun indicates approximately the true date of the treatise ? 

In our table, the time of no error of the sun is given as A. D. 250. 
The correctness of this date, however, is not to be too strongly insisted 
upon, being dependent upon the correctness with which the sun's place 
was first determined, and then referred to the point assumed as the 
origin of the sphere. It was, of course, impossible to observe directly 
when the sun's centre, by his mean motion, was 10' east of t Piscinm, 
and there are grave errors in the determination by the Hindus of the 
distances from that point of the other points fixed by them in their 
aodiac And a mistake of 1° in the determination of the sun's place 
would occasion a difference of 425 years in the resulting date of no 
error. We shall have occasion to recur to this subject m connection 
with the eighth chapter. 

There is also an alternative supposition to that which we have made 
above, respecting the conclusion from the date of no error of the sun. 
If the error in the sun's motion were a fundamental feature of the whole 
Hindu system, appealing alike in all the different text-books of the 
science, that date would point to the origin rather of the whole system 
than of any treatise which might exhibit it But although the different 
Siddbantas nearly agree with one another respecting the length of the 
sflereal year, they do not entirely accord, as is made evident by the 
following statement, in which are included all the authorities to which 
we have access, either in the original, or as reported by Colebrooke, 
Bpntley, and Warren : 



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168 



1 


JE. Burgess, efc., 


P. 


Authority. 


Lenftb of sidamal yew. 


Error. 


Sdrya-Siddhaoto, 


365d 6h i2«36«.56 


+ 3m 25^.8l 


Pauliga-Siddhanta, 


365 6 12 36 


+ 3 25.a5 


Para^ra-Siddhanta, 


365 6 12 3i.5o 


+ 3 20.75 


Arya-Siddhanta, 


365 6 12 3o.84 


-4- 3 20.09 


Laghu-Arys-Siddhanta, 


365 6 ia 3o 


+ 3 19.25 


SiddhAnta-^irotiuuju, 


365 6 12 9 


+ 3 58.25 



The first five of these might be regarded as unimportant variations 
of the same error, bat it would seem that the last is an independent de- 
termination, and one of later date than the others ; while, if all are 
independent, that of the Surya-Siddhanta has the appearance of being 
the most ancient Such questions as these, however, are not to be too 
hastily decided, nor from single indications merely ; they demand the 
most thorough investigation of each different treatise, and the careful 
collection of all the evidence which can be brought to bear upon them. 

Here lies Bentley's chief error. He relied solely upon his method of 
examining the elements, applying even that, as we have seen, only par- 
tially and uncritically, and never allowing his results to be controlled 
or corrected by evidence of any other character. He had, in fact, no 
philology, and he was deficient in sound critical judgment He thor- 
oughly misapprehended the character of the Hindu astronomical litera- 
ture, thinking it to be, in the main, a mass of forgeries framed for the 
purpose of deceiving the world respecting the antiquity of the Hindu 
people. Many of his most confident conclusions have already been 
overthrown by evidence of which not even he would venture to question 
the verity, and we are persuaded that but little of his work would stand 
the test of a thorough examination. 

The annexed table presents a comparison of the times of mean side- 
real revolution of the planets assumed by the Hindu astronomy, as rep- 
resented by two of its principal text-books, with those adopted by the 
great Greek astronomer, and tnose which modern science has established. 
The latter are, for the primary planets, from Le Verrier;* for the moon, 
from Nichol (Cyclopedia of the Physical Sciences, London : 1867). 
Those of Ptolemy are deduced from the mean daily rates of motion in 
longitude given by him in the Syntaxis, allowing for the movement of 
the equinox according to the false rate adopted by him, of 36" yearly. 

Comparative Table of the Sidereal Revolutions of the Planets. 



Planet. 


Surya-Siddhanta. 


Siddhanta-^lromani 


Ptolemy. 


Moderns. 


Sun, 
Mercury, 
Venus, 
Mara, 

Jupiter, 
Saturn, 
Moon: 

aid. rev. 

synod, rev. 

rev. of apsis, 
" " node, 


d h m • 
365 6i2 36.6 
87 23 16 22 3 

334 16 45 56.3 

686 23 56 s3.5 
4,333 7 4i44.4 
10,765 18 33 i3.6 

27 7 43 13.6 

29 12 44 2.8 

3,232 2 14 53.4 

6,794 9 35 454 


d ha* 

365 6 12 9.0 

87 23 16 4i.5 

224 16 45 1.9 

686 23 57 1.5 

4,33s 5 45 43.7 

io,765 19 33 56.5 

27 7 43 12. r 

29 i2 44 2.3 

3,232 17 37 6.0 

6,792 6 5 41.9 


d fa m t 

365|6 948.6 

87 33 16 4a-9 

334 16 5i 56.8 

686 33 3i 56.i 

4,332 18 9 io.5 

10,758 17 48 14.9 

27 7 43 1 2.1 

291244 3.3 

3,232 9 52 i3.6 

6,799 a3 * 8 394 


d h m ■ 
365 6 910.8 
87 23i5 43.9 
224 16 49 80 
686 23 3o4M 
4,332 14 a 8.6 

10,759 5 l6 32.2 

27 7 43 114 

291244 2.9 

3,232 1 3 48 29.6 

6,798 6 4i456 



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i. 40.] 



S&rya-Siddh&rda. 



169 



In the additional notes at the end of the work, we shall revert to the 
subject of these data, and of the light thrown by them upon the origin 
and age of the system. 

84 The number of risings of the asterisms, diminished by 

the number of the revolutions of each planet respectively, gives 
the number of risings of the planets in an Age. 
. 85. The number of lunar months is the difference between the 
number of revolutions of the sun and of the moon. If from it 
the number of solar months be subtracted, the remainder is the 
number of intercalary months. 

36. Take the civil days from the lunar, the remainder is the 
number of omitted lunar days (tithikshayd). From rising to 
rising of the sun are reckoned terrestrial civil days; 

87. Of these there are, in an Age, one billion, five hundred 
and seventy-seven million, nine hundred and seventeen thousand, 
eight hundred and twenty-eight ; of lunar days, one billion, six 
hundred and three million, and eighty; 

88. Of intercalary months, one million, five hundred and 
ninety-three thousand, three hundred and thirty-six ; of omitted 
lunar days, twenty-five million, eighty-two thousand, two hun- 
dred and fifty-two ; 

89. Of solar months, fifty-one million, eight hundred and forty 
thousand. The number of risings of the asterisms, diminished 
by that of the revolutions of the sun, gives the number of ter- 
restrial days. 

40. The intercalary months, the omitted lunar days, the side- 
real, lunar, and civil days — these, multiplied by a thousand, are 
the number of revolutions, eta, in an JEon. 

Hie data here given are combinations o£ and deductions from, those 
contained in the preceding passage (w. 29-84), For convenience of 
reference, we present them below in a tabular form. 



In 4,320,000 jeert. In 1,080,000 yean. 

I,582,237,8a8 395,550,457 

4,320,000 1,080,000 

*.577>9*7,828 394,47*457 



Sidereal days, 
deduct solar revolutions, 

Batumi, or civil days, 

Sidereal solar yean, 4,3 20,000 1,080,000 

multiply by no. of eolar months in a year, 12 12 

Solar mouths, 

Jloou't sidereal revolutions, 57,753,336 1 4438,334 

deduct eolar revolutions, 4,320,000 1,080,000 

8ynodical revolutions, lunar months, 53,433,336 i3,358,334 

deduct solar months, 5 1,840,000 12,960,000 

Intercalary months, 1,593,336 398,334 

vol. vi. 22_ - 



5i ,840,000 1 2,960,000 



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170 R Burgess, etc., [i.40- 

Ltmar month., 53,433,336 1 3,358,334 

multiply by no. of lunar days in a nootii, Jo 3o 



Lunar days, 1,603,000,080 400,750,020 

deduct civil days, 1,577,917,828 394479*457 

Omitted lunar days, 25,082,252 6>7o,563 

We add a few explanatory remarks respecting some of the terms em* 
ployed in this passage, or the divisions of time which they designate. 

The natural day, nycthemeron, is, for astronomical purposes, reckoned 
in the Surya-Siddhanta from midnight to midnight, and is of invariable 
length ; for the practical uses of life, the Hindus count it from sunrise 
to sunrise ; which would cause its duration to vary, in a latitude as high 
as our own, sometimes as much as two or three minutes. As above 
noticed, the system of Brahmagupta and some others reckon the astro- 
nomical day also from sunrise. 

For the lunar day, the lunar and solar month, and the general con- 
stitution of the year, see above, under verse 13. The lunar month, 
which is the one practically reckoned by, is named from the solar month 
in which it commences. An intercalation takes place when two lunar 
months begin in the same solar month : the former of the two is called 
an intercalary month (adhimdsa, or adhim&saka, " extra month"), of the 
same name as that which succeeds it. 

The term " omitted lunar day" (tiihikshaya^ u loss of a lunar day") 
is explained by the method adopted in the calendar, and in practice, of 
naming the days of the month. The civil day receives the name of the 
lunar day which ends in it ; but if two lunar days end in the same solar 
day, the former of them is reckoned as loss {kshaya) y and is omitted, the 
day being named from the other. 

41. The revolutions of the sun's apsis (manda), moving east- 
ward, in an J3on, are three hundred and eighty-seven ; of that 
of Mars, two hundred and four; of that of Mercury, three hun- 
dred and sixty-eight ; 

42. Of that of Jupiter, nine hundred ; of that of Venus, five 
hundred and thirty-five; of the apsis of Saturn, thirty-nine. 
Farther, the revolutions of the nodes, retrograde, are : 

43. Of that of Mars, two hundred and fourteen ; of that of 
Mercury, four hundred and eighty-eight; of that of Jupiter, one 
hundred and seventy-four ; of that of Venus, nine hundred and 
three ; 

44. Of the node of Saturn, the revolutions in an JSon are six 
hundred and sixty-two : the revolutions of the moon's apsis and 
node have been given here already. 

In illustration of the curious feature of the Hindu system of astronomy 
presented in this passage, we first give the annexed table ; which shows 
the number of revolutions in the j£on, or period of 4,320,000,000 years, 
assigned by the text to the apsis and node of each planet, the resulting 
time of revolution, the number of years which each would require to 



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i. 44.] 



S&n/a-Siddlidnta. 



171 



pass through an arc of one minute, and the position of each, according 
to the system, in 1850 ; the latter being reckoned in oar method, from 
the vernal equinox. Farther are added the actual positions for Jan. 1, 
1850, as given by Biot (Traite d' Astronomie, torn. v. 529) ; and finally, 
the errors of the positions as determined by this SiddhAnta, 

Table of Revolutions and Present Position of the Apsides and Nodes of 

the Planets. 



Phi*!, 


No. of 
itv. in 


Time of rerolution, 
io years. 


No. of yean 
tol'of 


Reraltiof 

position. 

A. D. I860. 


True 


Error of . 
Hindu 




an Man. 


motion. 


A. D. I860. 


position. 


Apndet: 


* 




1 


« 


t 


Sun, 


38 7 


11,162,790.7 


5i6.8 


95 4 


IOO 22 


- 5 16 


Mercury, 


368 


Il,739,l3o.4 


543.5 


238 i5 


255 7 


- 16 52 


Venus, 


535 


8,074,766.4 


373.8 


97 39 


309 24 


-211 45 


Mars, 


ao4 


21,176,470.6 


980^ 


147 49 


i53 18 


- 5 29 


Jupiter, 


900 


4,800,000.0 


222.2 


189 9 


191 55 


- 2 46 


Saturn, 


3 9 


110,769,230.8 


5128.2 


a54 24 


270 6 


- i5 42 


Nod*: 














Mercury* 


488 


8,852,459.0 


409.8 


38 27 


46 33 


- 8 6 


Venus, 


903 


4,784o53.2 


221.5 


77 26 


75 19 


+ 27 


Mars, 


214 


20,186,915.9 


934.6 


57 49 


48 23 


+ 9 26 


Jupiter, 


174 


24,827,586.2 


1 M9.4 


97 26 


98 54 


- 1 28 


Saturn, 


66a 


6,525,678.2 


302.I 


118 7 


112 22 


+ 5 45 



A mere inspection of this table is sufficient to show that the Hindu 
astronomers did not practically recognize any motion of the apsides and 
nodes of the planets ; since, even in the case of those to which they 
assigned the most rapid motion, two thousand years, at the least, would 
be required to produce such a change of place as they, with their im- 
perfect means of observation, would be able to detect 

This will, however, be made still more clearly apparent by the next 
following table, in which we give the positions ot the apsides and nodes 
as determined by four different text-books of the Hindu science, for the 
commencement of the Iron Age. 

Positions of the Apsides and Nodes of the Planets, according to Different 
Authorities, at the Commencement of the Iron Age, 3102 B. C. 



Planet. 



Apttidn: 
Son, 
Mercury, 

V 60118, 

Mars, 
Jupiter, 
Saturn, 
Nodes: 
Mercury, 
Venus, 
Mars, 
Jupiter, 
8aturn, 



Sdrya-Siddhanta. 



(w.) 
(175) 
066) 
(242) 

(9*) 

(4o7) 

(17) 



217 7 48(«9) 
7 10 19 12 
2 1939 ot 

4 9$7 36< 

5 21 o o( 
7 26 36 36 



(3oo-)3 10 37 12 



Siddhanta- 
(iromani 



(WY.) ■ • 



(i5i) 

<**) 

(,33) 

(3oo) 
(18) 



36(: 



2 17 45 

7 14 47 2 
2 21 2 10 

4 8 18 1 

5 22 i5 

8 20 53 3i 



(238-) o 21 20 53 
(4o8-)2 o 5 



(221-) o 20 52 48 

(409-) 2 o 1 48 \«t *~-/ - v « - \H~*- f - ~ — t~ 
(97-) 1 10 8 24(122-) o 21 59 46|(i36-) 1 10 19 12 
(79-) 2 19 44 24 (ao-) a 22 2 38 (44-) a 20 38 24 
w ^ a _ *- .» ( 2 67-)3 i3 23 3i '*«-* * ■*» '* « 



Arya- 
Siddhanta. 



(rt*.) • • • 
210) 2 17 45 
7 o i4 24 
o 17 *6 
A 3 5o 2 
36|(378) 5 22 48 o 
(16) 4 29 45 36 



(*54) 
(3oo) 
J36) 



4 f ( 



36( 



48(24 
4( 



(230-)O20 9 36 

(432-) 2 02848(408-) 



(283-) 3 10 48 o 



Paracara- 
Siddhaota. 



219) 

(162) 

o) 

i4 9 ) 
(448) 

(*4) 



2 17 45 36 
7 o 4o 19 
2 20 42 43 

4 2 43 26 

5 22 35 24 
7 28 14 52 



(296-) o 21 x 26 

2 o 5 2 

(112-) 1 9 336 

(87-) 2 21 43 12 

(2$8~) 3 10 26 24 



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172 R Burgess, etc., p. 44- 

The data of the Arya and Paracara Siddhantas, from which the posi- 
tions given in the table are calculated, are derived from Bentley (Hind. 
Ask pp. 139, 144). To each position is prefixed the number of com- 
pleted revolutions ; or, in the case of the nodes, of which the motion is 
retrograde, the number of whole revolutions of which each falls short 
by the amount expressed by its position. 

The almost universal disagreement of these four authorities with 
respect to the number of whole revolutions accomplished, and their 
general agreement as to the remainder, which determines the position,* 
prove that the Hindus had no idea of any motion of the apsides and 
nodes of the planets as an actual and observable phenomenon ; but, 
knowing that the moon's apsis and node moved, they fancied that the 
symmetry of the universe required that those of the other planets should 
move also ; and they constructed their systems accordingly. They held, 
too, as will be seen at the beginning of the second chapter, that the 
nodes and apsides, as well as the conjunctions (ftgkra), were beings, 
stationed in the heavens, and exercising a physical influence over their 
respective planets, and, as the conjunctions revolved, so must these also. 
In framing their systems, then, they assigned to these points such a 
number of revolutions in an JEon as should, without attributing to them 
any motion which admitted of detection, make their positions what they 
supposed them actually to be. The differences in respect to the number 
of revolutions were in part rendered necessary by the differences of other 
features of the systems ; thus, while that of the Siddhanta-Qiromani 
makes the planetary motions commence at the beginning of the ./Eon, 
by that of the Surya-Siddhanta they commence 17,064,000 years later 
(see above, v. 24), and by that of the Arya-Siddhanta, 3,024,000 years 
later (Bentley, Hind. Ast. p. 139) : in part, however, thejr are merely 
arbitrary ; for, although the Pai&cara-Siddhanta agrees with the Sid- 
dhantarQiromani as to the time of the beginning of things, its numbers 
of revolutions correspond only in two instances with those of the latter. 

It may be farther remarked, that the close accordance of the different 
astronomical systems in fixing the position of points which are so difficult 
of observation and deduction as the nodes and apsides, strongly indicates, 
either that the Hindus were remarkably accurate observers, and all 
arrived independently at a near approximation to the truth, or that some 
one of them was followed as an authority by the others, or that all alike 
derived their data from a common source, whether native or foreign. 
We reserve to the end of this work the discussion of these different 
possibilities, and the presentation of data which, may tend to settle the 
question between them. 

45. Now add together the time of the six Patriarchs (manu), 
with their respective twilights, and with the dawn at the com- 
mencement of the JSon (kalpa) ; farther, of the Patriarch Manu, 
son of Vivasvant> 

* It is altogether probable that, in the two cases where the Arya-Siddh&nta seems 
to disagree with the others, its data were either given incorrectly by Bentley's au- 
thority, or have been incorrectly reported by him. 



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L 60.] S&rya-SiddkAnta. 173 

46. The twenty-seven Ages (yuga) that are past, and likewise 
the present Golden Age (krta yuga)\ from their sum subtract the 
time of creation, already stated in terms of divine years, 

47. In solar years : the result is the time elapsed at the end of 
the Golden Age ; namely, one billion, nine hundred and fifty- 
three million, seven hundred and twenty thousand solar years. 

We have already presented this computation, in full, in the notes to 
verses 23 and 24. 

48. To this, add tlje number of years of the time since 
past 

As the Surya-Siddh&nta professes to have been revealed by the Sun 
about the end of the Golden Age, it is of course precluded from taking 
any notice of the divisions of time posterior to that period : there is 
nowhere in the treatise an allusion to any of the #ras which are actually 
made use of by the inhabitants of India in reckoning time, with the ex- 
ception of the cycle of sixty years, which, by its nature, is bound to no 
date or period {see below, v. 55). The astronomical era is the com- 
mencement of the Iron Age, the epoch, according to this Siddh&nta, of 
the last general conjunction of the planets; this coincides, as stated 
above (under vv. 20-34) with Feb. 18, 1612 J. P., or 3102 B. C. From 
that time will have elapsed, upon the eleventh of April/ 1859, the 
number of 4060 complete sidereal years of the Iron Age. The com- 
putation of the whole period, from the beginning of the present order 
of things, is then as follows : 

From end of creation to end of last Golden Age, 1,953,720,000 

Silver Age, 1,396,000 

Braxen Age, 864,000 

Of Iron Age, 4,960 2,164,960 

Total from end of creation to April, 1859, 1,955,884,960 

Since the S&rya-Siddh&nta, as will appear from the following verses, 
reckons by luni-solar years, it regards as the end of I. A. 4960 not 
the end of the solar sidereal year of that number, but that of the luni- 
solar year, which, by Hindu reckoning, is completed upon the third of 
the same month (see War^tala Sankalita, Table, p. xxxii). 

48 ... . EeduGe the sum to months, and add the months 
expired of the current year, beginning with the light half of 
Cajtra. 

49. Set the result down in two places; multiply it by the 
number of intercalary months, ana divide by that of solar 
months, and add to the last result the number of intercalary 
months thus found ; reduce the sum to days, and add the days 
expired of the current month ; 

60. Set the result down in two places ; multiply it by the 
number of omitted lunar days, and divide by that of lunar days ; 
subtract from the last result the number of omitted lunar days 



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174 



& Burgess, etc., 



[i.50- 



thufl obtained: the remainder is, at midnight, on the meridian of 
Lanka, 
51. The sum of days, in civil reckoning. . . . 

In these verses is taught the method of one of the most important 
and frequently recurring processes in Hindu Astronomy, the finding, 
namely, of the number of civil or natural days which have elapsed at 
any given date, reckoning either from the beginning of the present 
creation, or (see below, v. 56) from any required epoch since that time. 
In the modern technical language, the result is uniformly styled the 
ahargana, " sum of days ; " that precise term, however, does not once 
occur in the text of the Surya-Siddhanta : in the present passage we 
have dyugana, which means the same thing, and in verse 53 dinar&fi, 
u heap or quantity of days." 

The process will be best illustrated and explained by an example. 
Let it be required to find the sum of days to the beginning of Jan. 1, 
1860. # 

It is first necessary to know what date corresponds to this in Hindu 
reckoning. We have remarked above that the 4960th year of the Iron 
Age is completed in April, 1859 ; in order to exhibit the place in the 
next following year of the date required, and, at the same time, to pre- 
sent the names and succession of the months, which in this treatise are 
assumed as known, and are nowhere stated, we have constructed the 
following skeleton of a Hindu calendar for the year 4961 of the Iron 
Age. 



Lttni-tol&r Year. 



Solar Year. 


mouth. 


first day. 


(I A i960.) 




12. Cftitra, 


Mar. i3, t85o. 


(L A. 4961.) 




1. Vaicafcha, 


Apr. 12, do. 


2. Jyaiahtha, 


May r 3, do. 


3. AshAdha, 


June 14 do. 


4. £ravana, 


July r 5, do. 


6. Bhadrapada, Aug. 16, do. 


6. Acvina, 


Sept. 1 6, do. 


7. Karttika, 


Oct 16, do. 


8. Margaciroha, 


, Nov. 1 5, do 


9. Pauaha, 


Dee. 1 5, do. 


10. Magna, 


Jan. 1 3, i860. 


it. Phalguna, 


Feb. n, da 


12. Caitra, 


Mar. ia, da 



month. 


ftntday. 


(I. A. 4961.) 




1. Caitra, 


Apr. 4, i85a 


2. Vaic&kha, 


May 3, da 


3. Jyaiahtha, 


June 2, do. 


4. Ashadha, 


July 1, do. 


5. £ravana, 


July 3 1, do. 


6. Bhadrapada, Aug. 29, do. 


7. Acvina, 


Sept 28, do. 


8. KArttika, 


Oct. 27, do. 


9. M&rgaciraha, 


, Nov. 26, do. 


10. Pausha, 


Dec. 25, do. 


11. Magha, 


Jan. 24, i860. 


12. Phalguna, 


Feb. 22, do. 


(LA. 4o6x) 




1. Oftitra, 


Mar. 23, do. 



The names of the solar months are derived from the names of the 
asterisms (see below, chap, viii.) in which, at the time of their being first 
so designated, the moon was full during their continuance. The same 
names are transferred to the lunar months. Each lunar month is divided 
into two parts; the first, called the light half (jpukla paksha, "bright 



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i 52.] S&rya-Siddhdnia. 175 

side"), lasts from n«w moon to full moon, or while the moon is waxing ; 
the other, called the dark half (kfskna paksha, "black side"), lasts from 
foil moon to new moon, or while the moon is waning. 

The table shows that Jan. 1, 1860, is the eighth day of the tenth 
month of the 4961st year of the Iron Age. The time, then, for which 
we have to find the sum of days, is 1,955,884,960 y., 9 m., 7 d. 

Number of complete yean elapsed, 1,955,884,960 

multiply by number of solar months in a year, 1 2 

Number of months, 23,470,619,520 

add mouths elapsed of current year, 9 

Whole number of momths elapsed, 33,470,619,599 

Now a proportion is made : as the whole number of solar months 
in an Age is to the number of intercalary months in the same period, so 
is the number of months above found to that of the corresponding 
iatercalary months : or, # 

5i,84o,ooo : 1,593,336 : : 23^70,619,529 : 72i,384,7o3 + 

Whole number of months, as above, 33,470,619,529 

add intercalary months, 73i,384»7o3 

Whole number of lunar months, 34,193,004,232 

multiply by number of lunar days in a month, 3o 

Number of lunar days, 735,760,136,960 

add lunar days elapsed of current month, 7 

Whole number of lunar days elapsed, 735,760,136,967 

To reduce, again, the number of lunar days thus found to the corres- 
ponding number of solar days, a proportion is made, as before : as the 
whole number of lunar days in an Age is to the number of omitted lunar 
days in the same period, so is the number of lunar days in the period 
for which the sum of days is required to that of the corresponding 
omitted lunar days : or, 

i,6o3,ooo,o8o : 35,082,253 : : 735,760,136,967 : n^56,oi8,3o5 + 

Whole number of lunar days as above, 725,760,126,967 

deduct omitted lunar days, 1 1,356,018,305 

Total number of civil days from end of creation ) mwJJ ^j . A o c._ 
tobegianingof Jan. i t i860, \ 7i4Ao4 f i<*fi 7 2 

This, then, is the required sum of days, for the beginning of the year 
A.D. 1860, at midnight, upon the Hindu prime meridian. 

The firet use which we are instructed to make of the result thus ob- 
tained is an astrological one. 

51 From this may be found the lords of the day, the 

month, and the year, counting from the sun. If the number be 
divided by seven, the remainder marks the lord of the day, be* 
ginning; with the sun. 

52. L>ivide the same number by the number of days in a 
month and in a year, multiply the one quotient by two and the 



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176 XL Burgess, eic. y [i. 62. 

other by three, add one to each product, and divide by seven ; 
the remainders indicate the lords of the month and of the 
year. 

These verses explain the method of ascertaining, from the sum of days 
already found, the planet which is accounted to preside over the day, 
and also those under whose charge are placed the month and year in 
which that day occurs. 

To find the lord of the day is to find the day of the week, since the 
latter derives its name from the former. The week, with the names and 
succession of its days, is the same in India as with us, having been 
derived to both from a common source. The principle upon which the 
assignment of the days to their respective guardians was made has been 
handed down by ancient authors (see Ideler, Handbuch d. math. n. tech. 
Chronologie, i. 178, etc.), and is well known. It depends upon the 
division of the day into twenty-four hours, and the assignment of each 
of these in succession to the planets, in their natural order ; the day 
^>eing regarded as under the dominion of that planet to which its first 
hour belongs. Thus, the planets being set down in the order of their 
proximity to the earth, as determined by the ancient systems of aa» 
tronomy (for the Hindu, see below, xii. <&!■§§), beginning with the 
remotest, as follows : Saturn, Jupiter, Mars, sun, Venus, Mercury, moon, 
and the first hour of the twenty-four being assigned to the Sun, as chief 
of the planets, the second to Venus, etc., it will be found that the twenty- 
fifth hour, or the first of the second day, belongs to the moon ; the forty- 
ninth, or the first of the third day, to Mars, and so on. Thus is obtained 
a new arrangement of the planets, and this is the one in which this 
Siddhanta, when referring to them, always assumes them to stand (see, 
for instance, below, v. 70 ; ii. 35-37) : it has the convenient property 
that by it the sun and moon are separated from the other planets, from 
which they are by so many peculiarities distinguished. Upon this order 
depend the rules here given for ascertaining also the lords of the month 
and of the year. The latter, as appears both from the explanation of 
the commentator, and from the rules themselves, are no actual months 
and years, but periods of thirty and three hundred and sixty days, fol- 
lowing one another in uniform succession, and supposed to be placed, 
like the day, under the guardianship of the planets to whom belong 
their first subdivisions : thus the lord of the day is the lord of its first 
hour ; the lord of the month is the lord of its first day (and so of its 
first hour) ; the lord of the year is the lord of its first month (and so 
of its first day and hour). We give below this artificial arrangement 
of the planets, with the order in which they are found to succeed one 
another as lords of the periods of one, thirty, and three hundred and 
sixty days ; we add their natural order of succession, as lords of the 
hours ; and we farther prefix the ordinary names of the days, with their 
English equivalents. Other of the numerous names of the planets, it 
is to be remarked, may be put before the word vdra to form the name 
of the day : vdra itself means literally u successive time," or " turn,*! 
and is not used, so far as we are aware, in any other connection, to 
denote a day. 



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i 52.] S&rya-Siddfidnta. 177 



1 


8&rya*Siddfidnta. 




Rune of day. 


Prewidbtf Ptu>«L 


Soeeewioa, at Lord of 
day, month, year, boa 


Ilaviv&ra, 


Sunday, 


Sun, 


I I I I 


Somavftra, 


Monday, 


Moon, 


a 5 6 4 


Mangulavara, 


Tuesday, 


Man, 


3 a 4 7 


Badhav&ra, 


Wednesday, 


Mercury, 


4 6 a 3 


Guru vara, 


Thursday, 


Jupiter, 


5 3 7 6 


QakravSra, 


Friday, 


Venus, 


675a 


fanivAra, 


Saturday, 


Saturn, 


7 4 3 5 



At the first day of the subsistence of the present order of things is 
■apposed to have been a Sunday, it is only necessary to divide the sum 
of days by seven, and the remainder will be found, in the first column, 
opposite the name of the planet to which the required day belongs. 
Tnoa, taking the sum of days found above, adding to it one, for the first 
trf January itself and dividing by seven, we have : 
7)7*4,404^08,573 
102,057,7*9,796-1 

The first of January, 1860, accordingly, fells on a Sunday by Hindu 
reckoning, as by our own. 

On referring to the table, it will be seen that the lords of the months 
fellow one another at intervals of two places. To find, therefore, by a 
vamraary process, tiie lord of the month in which occurs any given day, 
first divide the sum of days by thirty ; the quotient, rejecting the re- 
mainder, is the number of months elapsed ; multiply this by two, that 
each month may push the succession forward two steps, add one for the 
current month, divide by seven in order to get rid of whole series, and 
the remainder is, in the column of lords of the day, the number of the 
regent of the month required. Thus : 

3 o)7t44a4,to8,57» 

a3,8i3,47o,285+ 

a 



47,6^94^570 

1 

7)47,626,940^71 



6^o3,848,652 -7 

The regent of the month in question is therefore Saturn. 

By a like process is found the lord of the year, saving that, as the 
lords of the year succeed one another at intervals of three places, the 
multiplication is by three instead of by two. Upon working out the 

Jrocess, it will be found that the final remainder is five, which designates 
upiter as the lord of the year at the given time. 
Excepting here and in the parallel passage xii. 77, 78, no reference is 
made in the Surya-Siddhanta to the week, or to the names of its days. 
Indeed, it is not correct to speak of the week at all in connection with 
India, for the Hindus do not seem ever to have regarded it as a division 
•of time, or a period to be reckoned by ; they knew only of a certain order 
of succession, in which the days were placed under the regency of the 
•even planets. And since, moreover, as remarked above (under w. 11, 
vol. vi. 23 



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ttl 



-RBwhgess } 0te.+ 



&.'«•* 



12), they fcetef mad* that division of the day into twenty-four hours 
Upon which the order of regency depends, it follows that the whole ays* 
tern was of foreign origin, and introduced into India along with other 
elements of the modern sciences of astronomy and astrology, to which 
it belonged. Its proper foundation, the lordship of the successive hours, 
is shown by the other passage lxii. 78) to have been also known to the 
Hindus; and the name by whicn the hours ar* there called (AefoWfya) 
indicates beyond a question, the source .whence they derived it 

> 58. Multiple the mim of days (dinardp) by the atnnbe* of 
revolutions of anjr planet, and divide by the number of civil 
days ; the result is the position of that planet, in virtue of it* 
mean motion, in revohations and parts of a revolution. 

By the number of revolutions and of civil days is meant, of course, 
their number, as stated above, in an Age. For " position of the planet," 
etc., the text has, according to its usual succinct mode of expression, 
simply u is the planet, in revolutions, etc." There is no word for u po- 
sition" or "place" in the vocabulary of this Siddhanta. 

This verse gives the method of finding the mean place of the planets 
si any given time for which the sum of days has been ascertained, by 
a simple proportion : as the number of civil days in a period is to the 
number of revolutions during the same period, so is the sum of days to 
the number of revolutions and parts of a revolution accomplished down 
to the given time. Thus, for the sun : 

i»577,9*7,8a8 : 4,3ao,ooo : : 714404,108,57a : 1055,884,960™ fr 17 48' 1" 

; The mean longitude of the sun, therefore, Jan. 1st, 1860, at midnight 
pn the meridian of TTjjayint, is 257° 48' 7". We have calculated in this 
manner the positions of all the planets, and of the moon's apsis and 
node — availing ourselves, however, of the permission given fcdow, in 
verse 56, and reckoning only from the last epoch of conjunction, the be* 
ginning of the Iron Age (from which time the sum of days is 1,811,945), 
and also employing the numbers afforded by the lesser period of 
1,080,000 years — and present the results in the following table. 

• . Mean Places */ th* Planets, Jan. 1st, I860, midnight, at UjjayinL 



;.«-*• 


According to the 
Suryn-Siddhftnta. 


The same 
corrected by the M/a. 




(»▼.) » • • " 


■ • $ >t 


Son, 


(4.960) 8 17 48 7 


6 17 48 7 


Mercury, 


(20,597) 4 1 5 i3 8 


4 8 36 16 


Veaua, : 


(8,o63) 10 ax 8 59 


10 16 XI 22 


Mart, 


(2,637) 5 24 17 36 


5 24 17 36 


Jupiter, 


(4i8) 2 26 7 


2 22 41 4* 


Satan, 


(168) 3 20 11 12 


3 25 8 5o 


Moon, 


(66,1x8) 11 i5 23 24 


n i5 23 24 


" *P«». 


(56o) 10 9 4a 26 


10 8 3 i3 


44 node, 


(26»9 24 2* 4 


9 S2 46 5x 



• The positions are given as deduced both from the numbers .«f 
revolutions stated in the text*, and from th'-atme as corrected by-ths 



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LW.j Sfoya-SiddhdntcL 17f 

bpa : prefixed are tke numbers of complete revolutions eceoinplished 
since the epoch. In the cases of the moon's apsis and node, however* 
it was necessary to employ the numbers of revolutions given for die 
whole Age, these not being divisible by four, and also to add to their 
ascertained amount of movement their longitude at the epoch (see below; 
under w. 57, 58). 

64. Thus alao are ascertained the places of the conjunction 
(tfghra) and apsis (mandocca) of each planet, which have been 
mentioned as moving eastward ; and in like manner of the nodes, 
which have a retrograde motion, subtracting the result from a 
whole circle. 

The places of the apsides and nodes have already been given above 
(under vv. 41-44), both for the commencement of the Iron Age, and 
for A.D. 1850. The place of the conjunctions of the three superior 
planets is, of course, the mean longitude of the sun. In the case of the 
inferior planets, the place of the conjunction is, in fact, the mean place of 
the planet itself in its proper orbit, and it is this which we have given for 
Mercury and Venus in the preceding table : while to the Hindu appre- 
hension, the mean place of those planets is the same with that of the sun. 

56. Multiply by twelve the past revolutions of Jupiter, add 
the signs or the current revolution, and divide by sixty ; the 
remainder marks the year of Jupiter's cycle, counting from 
Vijaya. 

This is the rule for finding the current year of the cycle of sixty 
years, which is in use throughout all India, And which is called the cycle 
Of Jupiter, because the length of its years is measured by the passage 
of that planet, by its mean motion, through one sign of the zodiac. 
According to the data given in the text of tnis Siddh&nta, the length of 
Jupiter's year is 361 d 0* 38 m ; the correction of the Uja makes it about 
12™ longer. It was doubtless on account of the near coincidence of 
this period with the true solar year that it was adopted as a measure of 
time ; but it has not been satisfactorily ascertained, so &r as we are 
aware, where the cycle originated, or what is its age, or why it was 
made to consist of sixty years, including five whole revolutions of the 
planet. There was, indeed, also in use a cycle of twelve of Jupiter's 
years, or the time of one sidereal revolution : see below, xiv. 17. Davis 
(As. Res. iii. 209; etc.) and Warren (K&la Sankalita, p. 107, e,tc.) have 
treated at some length of the greater cycle, and of the different modes 
of reckoning and naming ita yean riaual in the different province, of 
India. 

In illustration of the rule, let us ascertain the year of the cycle cor- 
responding to the present year, A.D. 1859. It is not necessary to make 
the calculation from the creation, as the role contemplates ; for, since 
the number of Jupiter's revolutions in the period of 1,080,000 years is 
divisible by five, a certain number of whole. cycles, without a remainder, 
will have elapsed at the beginning of the Iron Age. The revolutions of 
(he planet since that time, aa stated in the table test given, are 418, and 
it is in the 3rd sign of the 419th revolution ; the reduction Of ^the whote 



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180 



£ Burgess, ett. t 



fcts- 



amount ef movement to signs show* us thai the current year is the 
5019th since the epoch : divide this by 60, to cast out whole cycles, *Jui 
the remainder, 30, is the number of the year in the current cycle. This 
treatise nowhere rives the names of the years of Jupiter, but, as in the. 
case of the months, the signs of the zodiac, and otner similar matters, 
• assumes them to be already familiarly known in their succession : we 
accordingly present them below. We take them from Mr. Davis's paper, 
alluded to above, not having access at present to any original authority 
which contains them. 



i. Vjjaya. 
a. Jaya. 


ai. PramAdin. 
23. Anaoda. 


4r. Qrimukhs. 
42. Bh&va. 


3. Manmatha. 


a3. R&kshasa. 


43. Yuvan. 


4. Durmukha. 


a4. Anala. 


44 Dbatar. 


5. Hemabunba. 


25. Pingala. 


45. tjvara. 


6. Vflamba. 


26. Kalayukta, 


4& BabodhaDya, 


7. Vtkarm. 


27. Siddharthsx 


47. Pramathin. 


8. Oarvmrl 
* Am. 


28. ftaudra. 


48. Vikrama. 


29. DmmatL 
3ot Dnndnbhi. 


4^ Bbrcra. 
5o. Cittafchanu. 


ix. Oubhana. 


3i. RudhirodgAria, 


5i. Subb&niu 


ia. Krodhia, 


32. Raktaksba. 


52. Tarana. 


1 3. Vicvflvasu. 


33. Krodhana. 


53. Pftrthiva. 


14. Parabbava. 


34. Kshaya. 

35. Prabhara. 


54 Vyaya. 


1 5. Planing*. 


55. Sanrajit 


16. Ktiaka. 


36. Vibhava. 


56. Sarvadbarin. 


17. Saamya. 


37. 9okla. 

38. Pramoda. 


57. Virodhin. 


18. Sadbarana. 


58. Vikrta. 


19. Virodbakrt. 


391 PrajApatl 


59. Khara. 


20. Paridhflrhi. 


4o. Angtraa, 


60. Nandaaa. 



It appears, then, that the current year of Jupiter's cycle ia named 

Prajapati : upon dividing by the planet's mean daily motion the part of 

J»ke.current aign already passed over, it will be found that, according to 

"Ihe text, that year commenced on the twenty-third of February, 1859 ; 

or, if the correction of the btfa be admitted, on the third of April. 

Although it ia thus evident thai the S&rya-Siddhanta regards both 
the existing order of things and the Iron Age as having begun with 
Vyaya, that year is not generally accounted as the first, but as the 
twenty-seventh, of the cycle, which is thus made to commence with 
Prabhava* An explanation of this discrepancy might perhaps throw 
important light upon the origin or history of the cycle. 

This method of reckoning time is called (see below, xiv. 1, 2) the 
bdrhaspatya m&na, " measure of Jupiter." 

56. The processes which have thus been stated in frill detail, 
are practically applied in an abridged form. The calculation of 
the mean place of the planets may be made from any epoch 
(i/uga) that may be fixed upon. 

57. Now, at" the end of the Golden Age (krta yuffa), all the 
planets, by their mean motion — excepting, however, their node* 
and apsides (mandocca) — are in conjunction in the first of Ariea. 

58. The moon's apgis {uecd) is in the first of Capricorn, and 
its node is in the first of Libra ; and the rest, which have been 



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Its.} S6ty*8ddMnta. 181 

dated above to tare a alow motion— their position cannot be 
expressed in whole sign* 

It is curious to observe how the Surya-Siddhanta, lest it should seem 
to admit a later origin than that which it claims in the second verse of 
this chapter, is compelled to ignore the real astronomical epoch, the 
beginning of the Iron Age ; and also how it avoids any open recog- 
nition of the lesser cycle of 1,080,000 years, by which its calculations 
are so evidently intended to be made. 

The words at the end of verse 56 the commentator interprets to mean : 
"from the beginning of the current, i. e., the Silver, Age." In this he 
is only helping to keep up the pretence of the work to immemorial an- 
tiquity, even going therein beyond the text itself, which expressly says : 
" from any desired (ishtatas) yuga? Possibly, however, we have taken 
too great a liberty in rendering yuga by " epoch," and it should rather 
be * Age," i. e., w beginning of an Age." The word yuga comes from 
the root yuj, "to join n (Latin, jungo ; Greek, tetyrvtu : the word itself 
is the same with jugum, CvyoV), and seems to have been originally ap- 
plied to indicate a oycle, or period, by means of which the conjunction 
or correspondence of discordant modes of reckoning time was kept up ; 
thus it still signifies also the lu»trum, or cycle of five years, which, with 
an intercalated month, anciently maintained the correspondence of the 
year of 360 days with the true solar year. From such uses it was trans- 
ferred t6 designate the vaster periods of the Hindu chronology* 

As half an Age, or two of the lesser periods, are accounted to have 
elapsed between the end of the Golden and the beginning of the Iron 
Age, the planets, at the latter epoch, have again returned to a position 
of mean conjunction : the moon's node, also, is still in the first of Libra, 
.but her apsis has changed its place half a revolution, to the first of 
Cancer (see above, under w. 20-34). The positions of the apsides and; 
nodes of the other planets at the same time have been given already, 
under verses 41*44. 

The Hindu names of the signs correspond in signification with onr own, 
having been brought into India from the West There is nowhere in 
this work any allusion to them as constellations, or as having any fixed 
position of their own in the heavens : they are simply the names of the 
successive signs (rdfi, bka) into which any circle is divided, and it is left 
to be determined by the connection, in any ease, from what point they 
shall be counted. Here, of course, it is the initial point of the fixed 
Hindu sphere (see above, under v. 27). As the signs are, in the sequel, 
frequently cited by name, we present annexed, for the convenience of 
reference of these to whose memory they are not familiar in the order 
- of their succession, their names, Latin and Sanskrit, their numbers, and 
the figures generally used to represent them. Those enclosed in 
brackets do not chance to occur in our text. 

7. libra. £± tvld. 

8. Scorpio, Dl [vrfcika,] dli. 

9. Sagittarius, / dhanttt. 

10. Oapriooraus, VT makara, tnfffcu 

11. Aquariw, m htmbha. 
1 a. Pitcw, H [mina]. 



x. Atit% 


cp m*A4,qk 


a. Taurus, 


g frtAa» % 


3. Gemini, 


O mitkwMU 


4. Cancer, 


fS karka, karhxta. 


5. Lso, 


ft l9inka± 


6. Viigo. 


OB kangt 



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182 



E. Burgess, etc., 



[i58-: 



In the translation given above of the 'second half of verse 58, not a 
little violence is done to the natural construction. This would seem to 
require that it be rendered : " and the rest are in whole signs (have come 
to a position which is without a remainder of degrees) ; they, being pf 
slow motion, are not stated here." But the actual condition of things 
at the epoch renders necessary the former translation, which is that of 
the commentator also. We cannot avoid conjecturing that the natural 
rendering was perhaps the original one, and that a subsequent alteration 
of the elements of the treatise compelled the other and forced interpre- 
tation to be put upon the passage. 

The commentary gives the positions of the apsides and nodes (those 
of the nodes, however, in reverse) for the epoch of the end of the Golden 
Age, but, strangely enough, both in the printed edition and in our manu- 
script, commits the blunder of giving the position of Saturn's node a 
second time, for that of his apsis, ana also of making the seconds of the 
position of the node of Mars 12, instead of 24. We therefore add them 
below, in their correct form. 

Motion of the Apsides and Nodes of the Planets, to the End of the tost 

Golden Age, 



• PlUMt 


Ainu. 


Node. 




(rer.) ■ • # .• 


(rer.) ■ , ., 


Sun, 


(175) O 7 28 12 




Mercury, 


(166) 5 4 4 48 


(220) 8 II 16 48 


Venus, 


(a4i) 11 i3 ax 


(408) 4 17 a5 48* 


Ma», 


(9a) 3 3 14 a4 


(96) 9 11 20 24 


Jupiter, 


(407) 0900 


(78) 8 8 56 24 


Saturn, 


(17) 7 19 35 a4 


(299) 4 20 i3 12 



The method of finding the mean places of the planet* for midnight 
on the prime meridian having been now fully explained, the treatise 
proceeds to show how they may be found for other places, and for other 
times of the day. To this the first requisite is to know the dimensions 
of the earth. 

59. Twice eight hundred yqjanasxre the diameter of the earth :. 
the square root of ten times the square of that is the earth's cir- 
cumference. 

60. This, multiplied by the sine of the co-latitude {Jambajya\ 
of any place, ana divided by radius (trijtvd), is the corrected 
(sphuta) circumference of the earth at that place 

There is the same difficulty in the way of ascertaining the exactness 
of the Hindu measurement of the earth as of the Greek; the uncer- 
tain value, namely, of the unit of measure employed. The ycjana is 
ordinarily divided into ifcropo, "cries" (i. e., distances to which a certain; 
cry may be heard) ; the Jeropa into dhanus, " bow-lengths," or danda^ 
" poles ; " and these again into hasta, " cubits." By its origin, the latter 

— ■ ■■ ■ .... . i , ■'«*, 

* The printed edition, by an error of the pcm, gives 4. 



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i/eo| Sikya^Siddh&nia. 183 

6ught not to vary far from eighteen inches ; but the higher measures 
differ greatly in their relation to it The usual reckoning makes the 
yojana equal 32,000 cubits, but it is also sometimes regarded as com- 
posed of 16,000 cubits; and it is accordingly estimated oy different au- 
thorities at from four and a half to rather more than ten miles English. 
?nis uncertainty is no merely modern condition of things : Hiuen-Thsang, 
the Chinese monk who visited India in the middle of the seventh cen- 
tury, reports (see Stanislas Jnlien's Memoires de Hiouen-Thsang, i. 59, 
etc.) that in India "according to ancient tradition a yojana equals forty 
li; according to the customary use of the Indian kingdoms, it is thirty 
8; but the yojana mentioned in the sacred books contains only sixteen 
H : 9 this smallest yojana, according to the value of the li given by Wil- 
liams (Middle Kingdom, ii. 154), being equal to from five to six English 
miles. At the same time, Hiuen-Thsang states the subdivisions of the 
yojana in a manner to make it consist of only 16,000 cubits. Such 
being the condition of things, it is clearly impossible to appreciate the 
value of the Hindu estimate of the earth's dimensions, or to determine 
haw far the disagreement of the different astronomers on this pointmay 
be owing to the difference of their standards of measurement. Arya- 
bhatta (see Colebrooke's Hind. Alg. p. xxxviii; Essays, ii. 468) states the 
earth's diameter to be 1050 yojanas; Bhaskara (Siddh.-^ir. vii. 1) gives 
it as 1581 : the latter author, m his Lll&vatl (i. 5, 6), makes the yojana 
consist of 32,000 cubits. 

The ratio of the diameter to the circumference of a circle is here 
made to be 1 : ^10, or 1 : 3.1623, which is no very near approximation. 
It is not a little surprising to find this determination in the same treatise 
with the much more accurate one afforded by the table of sines given in 
the next chapter (w. 17-21), of 3488 : 10,800, or 1 : 3.14136 ; and then 
farther, to find the former, and not the latter, made use of in calculating 
the dimensions of the planetary orbits f see below, xiL 83). But the 
flame inconsistency is found also in other astronomical and mathematical 
authorities. Thus Aryabhatta (see Colebrooke, as above) calculates the 
earth's circumference from its diameter by the ratio 7 : 22, or 1 : 3.14286, 
trot makes the ratio 1 : */10 the basis of his table of sines, and Brahma- 
gupta and Qrldhara also adopt the latter. Bhaskara, in stating the 
earthV circumference at 4967 yojanas, is very near the truth, since 
1581 : 49*: : 1 : 8.14168 : his Llkvat! (v. 201) gives 7 : 22, and also, 
as more exact, 1250 : 3027, or 1 : 3.1416. This subject will be reverted 
to. in connection with the table of sines. 

The greatest circumference of the eartk as calculated according to 
the data and method of the text* is 5059.&W yojanas. The astronomical 
yojana must be regarded as an independent standard of measurement, 
by which to estimate the value of the otheT dimensions of the solar 
sytetem stated in this treatise. To make the earth's mean diameter cor- 
rect as. determined by the Surya-Siddhanta, the yojana should equal 
4.94 English miles ; to make the circumference correct, it should equal 
4b91 miles. 

T%e rule for finding the circumference of the earth upon a parallel of 
latitude is founded upon a simple proportion, viz., rad. : cos. latitude : : 
circ of earth 4fc equator : do, at the given parallel ; the cosine of the 



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184 JE Burgess, etc., p. fto- 

latitnde being, in effect, the radius of the circle of latitude. Radius and 
cosine of latitude are tabular numbers, derived from the table to be 
given afterward (see below, ii. 17-21). This treatise is not accustomed 
to employ cosines directly in its calculations, but has special names for 
the complements of the different arcs which it has occasion to use* 
Terrestrial latitude is styled aksha^ "axle," which term, as appears from 
xii. 42, is employed elliptically for eksKonnati, * elevation of the axle," 
i. e^ "of the pole : " lamba, co-latitude, whioh properly signifies " lap 
ging, dependence, foiling off," is accordingly the depression of the pole, 
or its distance from the sentth. Directions for finding the co4atitude 
are given below (iii. 13, 14). 

The latitude of Washington being 88° 54', the sine of its oo4atitud* 
is 2675'; the proportion 8488 : 2675 : : 5059.64 : 8936.75 gives us, then, 
the earth's circumference at Washington as 8986.75 ycjanas. 

60. . . . Multiply the daily motion of a planet by the distance 
in longitude (deginiara) of any place, and oivide by its corrected 
circumferenoe; 

61. The quotient, in minutes, subtract from the mean position 
of the planet as found, if the place be east of the prime meridian 
(rekhd) ; add, if it be west; the result is the planet's mean po* 
sition at the given place. 

The rules previously stated have ascertained the mean places of the 
planets at a riven midnight upon the prime meridian ; this teaches us 
now to find them for the same midnight upon any other meridian, or> 
how to correct for difference of longitude the mean places already found* 
The proportion is : as the circumference of the earth at the latitude of 
the point of observation is to the part of it intercepted between that 
point and the prime meridian, so is the whole daily motion of each 
planet to the amount of its motion during the time between midnight 
on the one meridian and on the other. The distance in longitude 
(isf&ntmra, literally u difference of region 9 ') is estimated, it will he ob- 
served, neither in time nor in arc, but in yojanas. How it is ascertained 
is taught below, in verses 63-65. 

The geographical position of the prime meridian (rskhA, literally 
u line ") is next stated. ^ 

62. Situated upon the line which passes through the haunt of 
the demons (rdkshasa) and the mountain which is the seat of the 
£odgy are Ronttaka and Avantt, as also the adjacent lake. 

The " haunt of the demons" is Lanka, the fabled seat of Havana, the 
chief of the Rakshasas, the abduction by whom of Rama's wife, with 
the expedition to Lanka of her heroic husband for her rescue, its ac- 
complishment, and the destruction of Havana and his people, form the 
subject of the epic poem called the Ramayana. In that poem, and to 
the general apprehension of the Hindus, Lanka is the island Ceylon ; in 
the astronomical geography, however (see below, xii. 89), it is a city, 
situated upon the equator. Hew far those who established the meridian 
may have regarded the actual position of Ceykm as identical with that 



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L62.] Sdrya-Skldltdnta. 186 

assigned to Lanka, might not be easy to determine. The "seat of the 
gods" is Mount Mem, situated at the north pole (see below, xii. 34, etc.). 
The meridian is usually styled that of Lanka, and " at Lank&" is the 
ordinary phrase made use of in this treatise (as, for instance, above, v. 
50; below, hi. 43) to designate a situation either of no longitude or of 
no latitude. 

But the circumstance which actually fixes the position of the prime 
meridian is the situation of the city of Ujjayini, the 0%qvn of the Greeks, 
the modern Ojein, It is called in the text by one of its ancient names, 
Avantl It is the capital of the rich and populous province of M&lava, 
occupying the plateau of the Vindhya mountains just north of the 
principal ridge and of tho river Narmada, (Nerbudda), and from old 
time a chief seat of Hindu literature, science, and arts. Of all the cen- 
tres of Hindu culture, it lay nearest to the great ocean-route by which, 
during the first three centuries of our era, so important a commerce was 
carried on between Alexandria, as the mart of Rome, and India and the 
countries lying still farther east That the prime meridian was made 
to pass through this city proves it to have been the cradle of the Hindu 
science of astronomy, or its principal seat during its early history. Its 
actual situation is stated by Warren (K&la Sankalita, p. 9) as lat 23° 
11' 30" N, long. 75° 58' E. from Greenwich : a later authority, Thorn- 
ton's Gazetteer of India (London : 1857), makes it to be in lat 23° 10' N., 
long. 75° 47' E. ; in our farther calculations, we shall assume the latter 
position to be the correct one. 

The situation of Rohltaka is not so clear; we have not succeeded ia 
finding such a place mentioned in any work on the ancient geography 
of India to which we have access, nor is it to be traced upon Lassen's 
knap of ancient India. A city called Rohtuk, however, is mentioned by 
Thornton (Gazetteer, p. 836), as the chief place of a modem British 
district at the same name, and its situation, a little to the north-west of 
Delhi, in the midst of the ancient Kurukshetra, leads us to regard it aa 
identical with the Rohltaka of the text. That the meridian of Lanka, 
was expressly recognized as passing over the Kurukshetra, the memora- 
ble site of the great battle described by the Mah&bharata, Beems clear. 
Bh&skara (Siddh.-Qir., Gan^ vii. 2) describes it as follows: "the line 
which, passing above Lanka, and Ujjayini, and touching the region of 
the Kurukshetra, etc., goes through Meru — that lmef)i by the wise 
regarded as the central meridian (madhyarekhd) of the earth." Our 
own commentary also explains sannihitam iarah, which we have transla- 
ted "adjacent lake," as signifying Kurukshetra. Warren (as above) 
takes the same expression to be the name of a city, which seems to us 
highly improbable ; nor do we see that the word saras can properly be 
applied to a tract of country : we have therefore thought it safest to 
translate literally the words of the text, confessing that we do not know 
to what they refer. 

If Rohltaka and Rohtuk signify the same place, we have here a 
measure of the accuracy of the Hindu determinations of longitude ; 
Thornton gives its longitude aa 76° 38', ot 51' to the east of Ujjayini. 

The method by which an observer ia to determine his distance from 
the prime meridian ia next explained. 
vol. vi. 94 



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186 E. Burgess, etc., [I G3- 

63. When, in a total eclipse of the moon, the emergence 
(unmilana) takes place after the calculated time for its occur- 
rence, then the place of the observer is to the east of the central 
meridian ; 

64. When it takes place before the calculated time, his place 
is to the west : the same thing may be ascertained likewise from 
the immersion (nimilana). Multiply by the difference of the two 
times in nfidis the corrected circumference of the earth at the 
place of observation, 

65. And -divide by sixty : the result, in yojanas, indicates the 
distance of the observer from the meridian, to the east or to the 
west, upon his own parallel ; and by means of that is made the 
^correction for difference of longitude. 

Choice is made, of course, of a lunar eclipse, and not of a solar, for 
the purpose of the determination of longitude, because its phenomena, 
being unaffected by parallax, are seen everywhere at the same instant of 
absolute time ; and the moments of total disappearance and first reap- 
pearance of the moon in a total eclipse are farther selected, because the 
precise instant of their occurrence is observable with more accuracy than 
that of (the first and last contact of the moon with the shadow. For 
the explanation of the terms here used see the chapters upon eclipses 
(below, iv-vi). 

The interval between the computed and observed time being ascer* 
tained, the distance in longitude (defdntara) is found by the simple 
proportion : as the whole number of n&dts in a day (sixty) is to the inter- 
val of time in nadis, so is the circumference of the earth at the latitude 
of the point of observation to the distance of that point from the prime 
meridian, measured on the parallel. Thus, for instance, the distance of 
Ujjayini from Greenwich, in time, being 5 h 3 m 8 s , and that of Washing- 
ton from Greenwich 5 h 8 m ll 8 (Am. Naut Almanac), that of Ujjayini 
jfrom Washington is 10 h ll m 19 8 , or, in Hindu time, 25 n 28 v 1*.8, or 
25M718 : And by the proportion 60 : 25.4718 : : 3936.75 : 1671.28, wa 
obtain 1671.28 yojanas as the distance in longitude {depdntara) of 
Washington from the Hindu meridian, the constant quantity to be em- 
ployed in finding the mean places of the planets at Washington. 

We might Wve expected that calculators so expert as the Hindu* 
would employ the interval of time directly in making the correction for 
difference of longitude, instead of reducing it first to its value in yojanas. 
That they did not measure longitude in our manner, in degrees, etc., is 
.owing to the fact that they seem never to have thought of applying: to 
the globe of the earth the system of measurement by circles and divi- 
sions of circles which they used for the sphere of the heavens, but, even 
when dividing the earth into zones (see below, xii. 59-66) reduced aU 
their distances laboriously to yojanas. 

66. The succession of the week-day (vdra) takes place, to the 
(east of the meridian, at a time after midnight equal to the differ* 
vonce of longitude in nadis ; to the west of the meridian, at a 
^corresponding time before midnight; 



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i 6Y.J S&rya-Siddhdnta. 187 

This verse appears to us to be an astrological precept, asserting the 
regency of the sun and the other planets, in their order, over the suc- 
cessive portions of time assigned to each, to begin everywhere at the 
same instant of absolute time, that of their true commencement upon 
the prime meridian ; so that, for instance, at Washington, Sunday, as 
the day placed under the guardianship of the sun, would really begin at 
eleven minutes before two on Saturday afternoon, by local time. The 
commentator, however, sees in it merely an intimation of what moment 
of local time, in places east and west of the meridian, corresponds to 
the true beginning of the day upon the prime meridian, and he is at 
much pains to defend the verse from the charge of being superfluous 
and unnecessary, to which it is indeed liable, if that be its only meaning. 

The rules thus far given have directed us only how to find the mean 
places of the planets at a given midnight. The following verse teaches 
the method of ascertaining their position at any required hour of the 
day. 

67. Multiply the mean daily motion of a planet by the number 
of nSdts of the time fixed upon, and divide by sixty : subtract 
the quotient from tbe place of the planet, if the time be before 
midnight ; add, if it be after : the result is its place at the given 
time. 

The proportion is as follows : as the number of nadis in a day (sixty) 
is to those in the interval between midnight and the time for which the 
mean place of the planet is sought, so is the whole daily motion of the 
planet to its motion during the interval ; and the result is additive or 
subtractive, of course, according as the time fixed upon is after or before 
midnight. 

In order to furnish a practical test of the accuracy of this text-book 
of astronomy, and of its ability to yield correct results at the present 
time, we have calculated, by the rule given in this verse, the mean longi- 
tudes of the planets for a time after midnight of the first of January, 
1860, on the meridian of Ujjayini, which is equal to the distance in 
time of the meridian of Washington, viz. 25 n 28 v 1P.8, or d .42453 ; and 
we present the results in the annexed table. The longitudes are given 
as reckoned from the vernal equinox of that date, which we make to be 
distant 18° 5' 8".25 from the point established by the Surya-Siddhanta 
as the beginning of the Hindu sidereal sphere ; this is (see below, chap, 
viii) 10' east of £ Piscium. We have ascertained the mean places both 
as determined by the text of our Siddhanta, and by the same with the 
correction of the bija. Added are the actual mean places at the time 
designated : those of the primary planets have been found from Le Ver- 
rier's elements, presented in Biot s treatise, as cited above ;* those of 
the moon, and of her apsis and node, were kindly furnished us from the 
office of the American Nautical Almanac, at Cambridge. 

* We would warn onr readers, however, of a serious error of the .press in the 
table as given by Biot; as the yearly motion of the earth, read 1,295,977.3$, instead 
of . . . 972.88. . . . 



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188 



Ei Burgess, etc., 



[U*- 



Afean Longitude* of the Planets, Jan. 1st, 1860, midnight, at Washington, 



Planet 


According to Surya-SWdhenta : 


According to 




text. 




with bija. 


moderns. 


Sun, 


• * 
96 18 


21 


« * «» 
96 18 21 


• 1 M 

100 5 6 


Mercury, 


i55 2 


3o 


148 25 39 


i5i 28 20 


Venus, 


33 9 54 


55 


334 57 18 


336 i3 36 


Mart, 


192 36 


5 


192 36 5 


197 26 32 


Jupiter, 


io4 7 


22 


too 48 56 


io3 35 17 


Saturn, 


128 17 


11 


i33 14 49 


137 10 10 


! Moon, 


9 4 


9 


9 4 9 


12 41 23 


'i " »!»», 


327 5o 


24 


326 11 11 


326 47 35 


" node, 


3i2 '29 


5i 


3io 5o 38 


3i2 48 10 



In the next following table is farther given a view of the errors of the 
Hindu determinations — both the absolute errors, as compared with the 
actual mean place of each planet, and the relative, as compared with 
the place of the sun, to which it is the aim of the Hindu astronomical 
systems to adapt the elements of the other planets. Annexed to each 
error is the approximate date at which it was nothing, or at which it 
will hereafter disappear, ascertained by dividing the amount of present 
error by the present yearly loss or gain, absolute or relative, of each 
planet ; excepting in the case of the moon, where we have made allow- 
ance, according to the formula used by the American Nautical Almanac, 
for the acceleration of her motion* 

Errors of the Mean Longitudes of the Planets, as calculated according to 
the Surya-Siddhanta. 





Errors according to text : | The seme, with btja .- 


Planet. 


absolute. 


when 
correct. 


ral. to ton. 


cSecV ***** • 


when 
correct. 


reL to mn. 


when 
correct. 




• 1 n 


A.D. 


' f 


X.D. 


1 a 


A,D. 


• t a 


A.D. 


Sun, 


-3 46 45 


25o 


OOO 


.... 


-3 4^ 45 


25o 


OOO 




Mercury, 


+3 34 10 


2332 


+7 20 55 


3271 


-3 2 41 


l5l7 


+o44 5 


1970 


Venus, 


+3 41 19 


1222 


+7 28 4 


9* 


-r 16 18 


2126 


+2 3o 27 


l5<?9 


Mars, 


-4 5o 27 


886 


-1 3 4a 


1455 


-4 5o 27 


886 


-1 3 4a 


1455 


Jupiter, 


+0 32 5 


1571 


+4 18 5o 


832 


-2 46 21 


4ao3 


+1 24 


1575 


Saturn, 


-8 52 55 


666 


-5 6 14 


85 7 


-3 55 21 


ia5o 


-0 8 36 


18*5 


Moon, 


-3 37 i4 


u5 


+0 9 3i 


1067 


-3 37 i4 


u5 


+0 9 3i 


1067 


" apsis, 


+1 a 49 


1679 


+4 49 34 


1262 


-0 36 24 


i#9 


+3 10 21 


1459. 


" node, 


-0 18 19 


1976 


+3 28 26 


1162 


-1 57 32 


2714 


+1 49 i3 


i466 



To complete the view of the planetary motions, and the statement trf 
the elements requisite for ascertaining their position in the sky, it only 
remains to give the movement in latitude of each, its deviation from th* 
general planetary path of the ecliptic. This is done in the concluding 
verses of the chapter. : 

68. The moon is, by its node, caused to deviate from the limit of 
its declination (krdnti), northward and southward, to a distance, 
when greatest, of an eightieth part of the minutes of a circle; 



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-^-X3- 9" 


or 


9/ X 10 as 90* as i° 30* 


^X4 = «C 


or 


ia' X 10 = iao 1 = a° 


9 


or 


tf X 10 = fry s= i° 


or 


ia' X 10 = 120' = a° 


=fx4-.* 


or 


it* X 10 sb iao' =b a° 



170.] Sarya^SiddMnt^ 189 

. 69. Jupiter, to the ninth part of that multiplied by two; 
Mars, to the same amount multiplied by three ; Mercury, Venus, 
and Saturn are by their nodes caused to deviate to the same 
amount multiplied by four. 

70. So also, twenty-seven, nine, twelve, six, twelve, and twelve, 
multiplied respectively by ten, give the number of minutes of 
mean latitude {yikshepayW the moon and the rest, in their order. 

The deviation of the planets from the plane of the ecliptic is here 
stated in two different ways, which give, however, the same results ; 
thus: •• 

Moon, ^^ = 370* or vf X 10 =* 370* = 4° 3o' 

Mara, 

Mercury, 

Jupiter, 

Venus, 

Saturn, 

The subject of the latitude of the planets is completed in verses 6-8, 
and verse 57, of the following chapter; the former passage describes 
the manner, and indicates the direction, in which the node produces its 
disturbing effect ; the latter gives the rule for calculating the apparent 
latitude of a planet at any point in its revolution. 

There is a little discrepancy between the two specifications presented 
in these verses, as regards the description of the quantities specified : 
the one states them to be the amounts of greatest (parama) deviation 
from the ecliptic ; the other, of mean (madhya) deviation* Both de- 
scriptions are also somewhat inaccurate. The first is correct only with 
reference to the moon, and the two terms require to be combined, in 
order to be made applicable to the other planets. The moon has its 
greatest latitude at 90° from its node, and this latitude is obviously 
equal to the inclination of its orbit to the ecliptic ; for although its 
absolute distance from the ecliptic at this point of its course vanes, as 
does its distance from the earth, on account of the eccentricity of its 
orbit, and the varying relation of the line of its apsides to that of its 
nodes, its angular distance remains unchanged. So, to an observer sta- 
tioned at the sun, the greatest latitude of any one of the primary planets 
would be the same in its successive revolutions from node to node, 
and equal to the inclination of its orbit But its greatest latitude as 
seen from the earth is very different in different revolutions, both on 
.account of the difference of its absolute distance from the ecliptic 
when at the point of greatest removal from it in the two halves of its 
orbit, and, much more, on account of its varying distance from the earth. 
The former of these two causes of variation jw&a- not recognized by the 



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190 



E. Burge*^ etc*, 



Q.70- 



Hindus : in this treatise, at least, the distance of the node from the 
apsis (mandocca) is not introduced as an dement into the process for 
determining a planet's latitude. The other cause of variation is duly 
allowed for (see below, ii. 57). Its effect, in the case of the three supe- 
rior planets, is to make their greatest latitude sometimes greater, and 
sometimes less, than the inclination of their orbits, according as the 
planet is nearer to us than to the sun, or thftcontrary ; hence the values 
given in the text for Mars, Jupiter, and Saturn, as they represent the 
mean apparent values, as latitude, of the greatest distance of each planet 
from the ecliptic, should nearly equal the inclination. In the case of 
Mercury and Venus, also, the quantities stated are the mean of the differ- 
ent apparent values of the greatest heliocentric latitude* but this mean 
is of course less, and for Mercury very much less, than the inclination. 
Ptolemy, in the elaborate discussion of the theory of the latitude con- 
tained in the thirteenth book of his Syntaxis, has deduced the actual 
inclination of the orbits of the two inferior planets : this the Hindus do 
not eeem to have attempted. 

We present below a comparative table of the inclinations of the 
orbits of the planets as determined by Ptolemy and by modern astrono- 
mers, with those of the Hindus, so far as given directly by the Surya- 
Siddhanta, 

Inclination of the Orbits of the Planets, according to Different Authorities. 



Planet 


Soiys-SiddbAnU. 


Ptolemy. 


Moderat. 


fiilli 


• 

i 3o 

i 

2 

4 3o 


• 

7 

3 3o 
i 

i 3o 
a 3o 
5 


• » »« 
708 
3 a3 3i 
r 5i 5 
1 18 4o 
a 29 28 
5 8 40 



The verb in verses 68 and 69, which we have translated " caused to 
deviate," is vi kshipyate, literally " is hurled away," disjicitur ; from it is 
derived the term used in this treatise to signify celestial latitude, vikshe- 
pa, "disjection." The Hindus measure tne latitude, however, as we 
shall have occasion to notice more particularly hereafter, upon a circle 
of declination, and not upon a secondary to the ecliptic. In the words 
chosen to designate it is seen the influence of the theory of the nodeV 
action, as stated in the first verses of the next chapter. The forcible 
removal is from the point of declination (kr&nti, " gait," or apahrama, 
" withdrawal," i. e., from the celestial equator) which the planet ought 
at the time to occupy. 

The title given to this first chapter (adhikdra, " subject, heading") is 
madhyam&dhik&ra, which we have represented in the title by "mean 
motions of the planets," although it would be more accurately rendered 
by " mean places of the planets ;" that is to say, the data and methods 
requisite for ascertaining their mean places. Now follows the spashta- 
dhikara, " chapter of the true, or corrected, places of the planets," 



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ii $.] SAryorSiddhdrda. 191 

* CHAPTER II. 

OF THE TRUE PLACES OF THE PLANETS. 

Otarr*fn >— 1-8, causes of the irregularities of the planetary motions ; 4-5, disturb- 
ing ioflnonee of the apsis and conjunction ; 6-8, of the node ; 9-1 1 , different 
degree of irregularity of the motion of the different planets; 12-13, different 
kinds of planetary motion; 14, purpose of this chapter; 16-16, rule for con- 
structing the table of sines; 17-22, table of smee; 22-27, table of veresd sines; 
tS, inclination of the ecliptic, and rule for finding the declination of any point in 
it; 29-80, to find the sine and cosine of the anomaly; 81-82, to find, by interpo- 
lation, the sine or versed sine corr es ponding to any given arc ; 88, to find, in like 
manner, the are corresponding to a given- sine or versed sine; 84-37, dimensions 
of the epicycles of the planets ; 88, to find the true dimensions of the epicycle at 
any point in the orbit; 89, to find the equation of the apsis, or of the centre; 
40-42, to find the equation of the conjunction, or the annual equation ; 48-45, 
application of these equations in finding the true places of the different planets ; 
48, correction of the place of a planet for difference between mean and apparent 
solar time; 47-49, bow to eonrect the daily motion of the planets for the effect of 
the apsis; 50-51, the same for that of the conjunction; 51-65, retrogradation of 
. the lesser planets ; 66, correction of the place of the node ; 57-58, to find the celes- 
tial latitude of a planet, and its declination as affected by latitude ; 59, to find the 
length of the day of any planet; 60, to find the radius of tho diurnal circle; 
61-68, to find the daw~sihe, and the respective length of the day and night ; 64, 
to find the number of asterisms traversed by a planet, and of days elapsed, since 
the commencement of the current revolution; 65, to find the yoga; 66, to find 
the current lunar day, and the time in it of a given instant ; 67-69, of the divisions 
of the lunar month called kara^a. 

1. Forms of Time, of invisible shape, stationed in the zodiac 
(bkagana), called the conjunction (ifyhrocca), apsis (mandocca)^ 
and node (pdta), are causes of the motion of the planets. 

2. The planets, attached to these beings by cords of air, are 
drawn away by them, with the right and left hand, forward or 
backward, according to nearness, toward their own place. 

3. A wind, moreover, called pravector (pravaha) impels them 
toward their own apices (ucca) ; being drawn away forward and 
backward, they proceed by a varying motion. 

4. The so-called apex (ticca), when in the half-orbit in front of 
the planety draws the planet forward ; in like manner, when in 
the half-orbit behind the planet, it draws it backward. 

6. When the planets, drawn away by their apices (ucca), move 
forward in their orbits, the amount of the motion so caused is 
called their excess (dhana) ; when they move backward, it is 
called their deficiency (rna). 

in these verses is laid before us the Hindu theory of the general 
nature of the forces which, produce the irregularities of the apparent 



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192 E. Burgess, etc., pi. 6. 

motions, regarded as being the real motions, of the planets. The world- 
wide difference between the spirit of the Hindu astronomy and that e# 
the Greek is not less apparent here than in the manner of presentation 
of the elements in the last chapter : the one is purely scientific, devis- 
ing methods for representing and calculating the observed motions, and 
attempting nothing farther ; the other is not content without fabricating 
a fantastic and absurd theory respecting the superhuman powers which 
occasion the movements with which it is dealing. The Hindu method 
has this convenient peculiarity, that it absolves from all necessity of 
adapting the disturbing forces to one another, and making them form 
one consistent system, capable of geometrical representation and mathe- 
matical demonstration ; it regards the planets as actually moving in 
circular orbits, and the whole apparatus of epicycles, given later in 
the chapter, as only a device for estimating the amount of the force, 
and of its resulting motion, exerted at any given point by the disturb- 
ing cause. 

The commentator gives two different explanations of the provector 
wind, spoken of in the third verse : one, that it is the general current, 
mentioned below, in xii. 73, as impelling the whole firmament of stars, 
and which, though itself moving westward, drives the planets, in some 
unexplained way, towards its own apex of motion, in the east ; the 
other, that a separate vortex for each planet, called provector on account 
of its analogy with that general current, although not moving in the 
same direction, carries them around in their orbits from west to east, 
leaving only the irregularities of their motion to be produced by the 
disturbing forces. This latter we regard as the proper meaning of the 
text : neither is very consistent with the theory of the lagging behind 
of the planets, given above, in i. 25, 26, as the explanation of their 
apparent eastward motion. The commentary also states more explicitly 
the method of production of the disturbance : a cord of air, equal in 
length to the orbit of each planet less the disk of the latter itself is 
attached to the extremities of its diameter, and passes through the two 
hands of the being stationed at the point of disturbance ; and he always 
draws it toward himself by the shorter of the two parts of the cord. 
The term ucca, which we have translated " apex," applies both to the 
apsis (manda, mandocca, "apex of slowest motion" — the apogee in the 
case of the sun and moon, the aphelion, though not recognized as such, 
in the case of the other planets), and to the conjunction (ftghra, ptgh- 
rocca, "apex of swiftest motion"). The statement made of the like 
effect of the two upon the motion of the planet is liable to cause diffi- 
culty, if it be not distinctly kept in mind that the Hindus understand 
by the influence of the disturbing cause, not its acceleration and retarda- 
tion of the rate of the planet's motion, but its effect in giving to the 
planet a position in advance of, or behind, its mean place. It may be 
well, for the sake of aiding some of our readers to form a clearer appre- 
hension of the Hindu view of the planetary motions, to expand and 
illustrate a little this statement of the effect upon them of the two 
principal disturbing forces. 

First, as regards the apsis. This is the remoter extremity of the major 
axis of the planet's proper orbit, and the point of its slowest motion. 



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ii.S.] S&rya-Skldhdnta. 193 

Upon passing this point, the planet begins to fall behind its mean place, 
but at the same time to gain velocity, so that at the quadrature it is 
farthest behind, but is moving at its mean rate ; during the next quad- 
rant it gains both in rate of motion and in place, until at the perigee, or 
.perihelion, it is moving most rapidly, and has made up what it before 
lost* so that the mean and true places coincide. Upon passing that point 
again, it gains upon its mean place during the first quafesnt, and loses 
what it thus gained during the second, until mean and true place again 
coincide at the apsis. Thus the equation of motion is greatest at the 
apsides, and nothing at the quadratures, while the equation of place is 
greatest at the quadratures, and nothing at the apsides; and thus the 
planet is always behind its mean place while passing from the higher to 
the lower apsis, and always in advance of it while passing from the 
lower to the higher; that is, it is constantly drawn away from its mean 
place toward the higher apsis, mandoeca. 

In treating of the effect of the conjunction, the ftghrocca, we have to 
-distinguish two kinds of cases. With Mercury and Venus (see above, 
i. 29, 31, 82), the revolution of the conjunction takes the place, in the 
Hindu system as in the Greek, of that of the planet itself the conjunc- 
tion being regarded as making the circuit of the zodiac in the same 
time, and in the same direction, as the planet really revolves about the 
eun ; while the mean place of these planets is always that of the sun 
itselfc While, therefore, die conjunction is making the half-tour of the 
heavens eastward from the sun, the planet is making its eastward elon- 
gation and returning to the sun again, being all the time in advance 
of its mean place, the sun ; when the conjunction reaches a point in the 
heavens opposite to the sun, the planet is in its inferior conjunction, or 
at he mean place; during the other half of the revolution of the con- 
junction, when it is nearest the planet upon the western side, the latter 
is making and losing its western elongation, or is behind its mean place. 
Accordingly, as stated in the text, the planet is constantly drawn away 
from its mean place, the sun, toward that side of the heavens in which 
the conjunction is. 

Once more, as concerns the superior planets. The revolutions as- 
signed to these by the Hindus are their true revolutions ; their mean 
places are their mean heliocentric longitudes ; and the place of the con- 
junction (ftghrocca) of each is the mean place of the sun. Since they 
move but slowly, as compared with the sun, it is their conjunction 
which approaches, overtakes, and passes them, and not they the con- 
junction. Their time of slowest motion is when in opposition with the 
sun ; of swiftest, when in conjunction with him : from opposition on to 
conjunction, therefore, or while the sun is approaching them from be- 
hind, they are, with constantly increasing velocity of motion, all the 
while behind their mean places, or drawn away from them in the direc- 
tion of the sun ; but no sooner has the sun overtaken and passed them, 
than they, leaving with their most rapid motion the point of coinci- 
dence between mean and true place, are at once in advance, and con- 
tinue to be so until opposition is reached again ; that is to say, they 
are still drawn away from their mean plaoe in the direction of the 

conjunction. . 

vol. vi. 25 



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194 E. Burgess, etc., pi. 5- 

The words used in verse 5 for "excess" and "deficiency," or for addi- 
tive and subtractive equation, mean literally "wealth" (dhana) and 
"debt" (rna). 

6. In like manner, also, the node, Kahu, by its proper force, 
caijses the deviation in latitude (vikshepa) of the moon and the 
other planetj, northward and southwajxl, fr° m their point of 
declination (apakrama). 

7. When in the half-orbit behind the planet, the node causes 
it to deviate northward ; when in the hair-orbit in front, it draws 
it away southward. 

8. In the case of Mercury and Venus, however, when the 
node is thus situated with regard to the conjunction (tfghra), 
these two planets are caused to deviate in latitude, in the manner 
stated, by the attraction exercised by the node upon the con- 
junction. 

The name Rahu, by which the ascending node is here designated, is 
properly mythological, and belongs to the monster in the heavens, which, 
by the ancient Hindus, as by more than one other people, was believed 
to occasion the eclipses of the sun and moon by attempting to devour 
them. The word which we have translated "force" is ranhas, more 
properly " rapidity, violent motion : " in employing it here, the text evi- 
dently intends to suggest an etymology for rdhu, as coming from the root 
rah or rank, " to rush on " : with this same root Weber (Lid. Stud. i. 
272) has connected the group of words in which rdhu seems to belong. 
For the Hindu fable respecting Rahu, see Wilson's Vishnu Purana, p. IS. 
The moon's descending node was also personified in a similar way, under 
the name of Ketu, but to this no reference is made in the present treatise. 

The description of the effect of the node upon the movement of the 
planet is to be understood, in a manner analogous with that of the effect 
of the apices in the next preceding passage, as referring to the direction 
in which the planet is made to deviate from the ecliptic, and not to that 
in which it is moving with reference to the ecliptic From the ascending 
node around to the descending, of course, or while the node is nearest to 
the planet from behind, the latitude is northern ; in the other half of the 
revolution it is southern. 

For an explanation of some of the terms used here, see the note to the 
last passage of the preceding chapter. 

As, in the case of Mercury and Venus, the revolution of the conjunc- 
tion takes the place of that of the planet itself in its orbit, it is necessary, 
in order to give the node its proper effect, that it be made to exercise 
its influence upon the planet through the conjunction. The commen- 
tator gives himself here not a little trouble, in the attempt to show why 
Mercury and Venus should in this respect constitute an exception to the 
general rule, but without being able to make out a very plausible case. 

9. Owing to the greatness of its orb, the sun is drawn away 
only a very little ; the moon, by reason of the smallness of its 
orb, is drawn away much more ; 



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ii. 14.] S&rya-Siddkdnta. 195 

10. Mars and the rest, on account of their small size, are, by 
the supernatural beings (ddivata) called conjunction (<fighrocca) 
and apsis (mandocca), drawn away very far, being caused to 
vacillate exceedingly. 

11. Hence the excess (dhana) and deficiency (rna) of these 
latter is very great, according to their rate of motion. Thus do 
the planets, attracted by those beings, move in the firmament, 
earned on by the wind. 

The dimensions of the sun and moon are stated below, in iv. 1 ; those 
of the other planets, in vii. 13. 

We have ventured to translate ativegita, at the end of the tenth verse, 
as it is given above, because that translation seemed so much better to 
suit the requirements of the sense than the better-supported rendering 
" caused to move with exceeding velocity." In so doing, we have assumed 
that the noun vega, of which the word in question is a denominative, re- 
tains something of the proper meaning of the root vij, " to tremble " 
from which it comes. 

12. The motion of the planets is of eight kinds : retrograde 
(vakra), somewhat retrograde (anuvo&ra), transverse (Jcutila), 
slow (manda), very slow (mandatara), even (sama) ; also, very 
swift (tfghratara), and swift (tfghra). 

13. Of these, the very swift (atifighra), that called swift, the 
slow, the very slow, the even — all these five are forms of the 
motion called direct (rju) ; the somewhat retrograde is retrograde. 

This minute classification of the phases of a planet's motion is quite 

fratuitous, so far as this Siddhanta is concerned, for the terms here given 
o not once occur afterward in the text, with the single exception of 
vdkra y which, with its derivatives, is in not infrequent use to designate 
retrogradation. Nor does the commentary take the trouble to explain 
the precise differences of the kinds of motion specified. According to 
Mr. Hoisington (Oriental Astronomer [Tamil and English], Jaffna: 1848, 
p. 133), anuvakra is applied to the motion of a planet, when, in retro- 
grading, it passes into a preceding sign. From the classification given in 
the second of the two verses it will be noticed that kutila is omitted : ac- 
cording to the commentator, it is meant to be included among the forms 
of retrograde motion ; we have conjectured, however, that it might possi- 
bly be used to designate the motion of a planet when, being for the 
moment stationary in respect to longitude, and accordingly neither ad- 
vancing nor retrograding, it is changing its latitude ; and we have trans- 
lated the word accordingly. 

14. By reason of this and that rate of motion, from day to 
day, the planets thus come to an accordance with their observed 
places (drg) — this, their correction (sphutikarana), I shall care- 
fully explain. 

Having now disposed of matters of general theory and preliminary 
explanation, the proper subject of this chapter, the calculation of the true 
(sphuia) from the mean places of the different planets, is ready to be 



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196 IS. Burgm, etc.) [ii. 14- 

token up. And the first thing in order is the table of sines, by means of 
which ail the after calculations are performed. 

15. The eighth part of the minutes of a sign is called the first 
sine (jydrdha); that, increased by the remainder left after sub- 
tracting from it the quotient arising from dividing it by itself) is 
the second sine. 

16. Thus, dividing the tabular sines in succession by the first, 
and adding to them, in each case, what is left after subtracting 
the quotients from the first, the result is twenty-four tabular 
sines (Jydrdhapinda), in order, as follows : 

17. Two hundred and twenty-five; four hundred and forty- 
nine ; six hundred and seventy-one ; eight hundred and ninety ; 
eleven hundred and five; thirteen hundred and fifteen; 

18. Fifteen hundred and twenty; seventeen hundred and nine- 
teen ; nineteen hundred and ten ; two thousand and ninety-three ; 

19. Two thousand two hundred and sixty-seven ; two thous- 
and four hundred and thirty-one; two thousand five hundred 
and eighty-five ; two thousand seven hundred and twenty-eight ; 

20. Two thousand eight hundred and fifty-nine ; two thousand 
nine hundred and seventy-eight: three tnousand and eighty- 
four ; three thousand one hundred and seventy-seven ; 

21. Three thousand two hundred and fifty-six ; three thousand 
three hundred and twenty-one ; three thousand three hundred 
and seventy-two ; three thousand four hundred and nine ; 

22. Three thousand four hundred and thirty-one ; three thous- 
and four hundred and thirty-eight. Subtracting these, in re- 
versed order, from the half-diameter, gives the tabular versed- 
sines (utJcramajydrdhapindaka) : 

23. Seven ; twenty-nine ; sixty-six ; one hundred and seven- 
teen ; one hundred and eighty -two ; two hundred and sixty -one ; 
three hundred and fifty-four ; 

24. Four hundred and sixty ; five hundred and seventy-nine ; 
seven hundred and ten; eight hundred and fifty-three; one 
thousand and seven ; eleven hundred and seventy-one; 

25. Thirteen hundred and forty-five; fifteen hundred and 
twenty -eight; seventeen hundred and nineteen; nineteen hund- 
red and eighteen ; 

26. Two thousand one hundred and twenty-three; two thous- 
and three hundred and thirty-three ; two thousand five hundred 
and forty -eight ; two thousand seven hundred and sixty-seven ; 

27. Two thousand nine hundred and eighty-nine ; three thous- 
and two hundred and thirteen ; three thousand four hundred and 
thirty-eight : these are the versed sines. 

We first present, in the following table, in a form convenient for refer- 
ence and use, the Hindu sines and versed sines, with the arcs to which 
they belong, the latter expressed both in minutes and in degrees and 
minutes. To facilitate the practical use of the table in making calcula- 



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iL2Y.] 



S&rya-Siddhdnta* 



197 



tions after the Hindu method, we hare added a column of the differences 
of the sines, and have farther turned the sines themselves into decimal 
parts of the radius. For the purpose of illustrating the accuracy of the 
table, we have also annexed the true values of the sines, in minutes, as 
found by our modern tables. Comparison may also be made of the deci- 
mal column with the corresponding values given in our ordinary tables 
of natural sines. 







Table 


of Sines and Versed Sines, 




No. 


Arcs, 


Hindu Sines, 


Trae Sine*, 


Verted 

Sines. 




in • ' 


in' 


in ' 


Dili: 


in parti of rad. 


in ' 


in' 


I 


3T W 


335' 


335' 


324' 


.065445 


334'.84 


7' 


3 


7' 3o' 


4S& 


449* 


322' 


.130599 


448'. 7 3 


39' 


3 


11° 15* 


675' 


671' 


2IQ* 

SIS' 


.195173 


67o'.67 


6& 


4 


15* 


ooc/ 


89c/ 


.358871 


889'. 7 6 


ur 


5 


i8 # 45' 


1 1 35' 


no5' 


310' 


.3ai4o8 


uo5'.o3 


182' 


6 


33* 3o' 


I350 1 


i3i5' 


3o5' 


.383489 


i3i5'.57+ 


361' 


7 


36* i5* 


i575' 


1 5 at/ 


199 1 

191' 
i83' 


442117 


i53o'.48 


354' 


8 


3o» 


1800* 


I7IQ* 


.5ooooo 


I7i8'.88 


460' 


9 


33*45' 


3035* 


1910* 


.555555 


IOOO*^! 


579 1 


10 


3/ 3o* 


335o* 


2093' 


174' 
164' 


.608784 


3093'.77 


710* 


ii 


4i # 1 5* 


24 7 5' 


3367* 


.659395 


126&.67 


853' 


13 


45° 


2700' 


343i' 


1 54' 


.707097 


343o'.86 


1007' 


i3 


48-45' 


3935* 


3585' 


i43' 


.751894 


3584'.64 


1171' 


i4 


53" 3o' 


3i5& 


3738' 


i3i' 


.793484 


3737'.35- 


i345* 


i5 


56 - iS' 


3375 1 


3859 1 * 


H9* 


.83i588 


3858'.38- 


i5a8' 


16 


&>• 


36<xy 


3978' 


.866301 


3977'.i8- 


1719/ 


17 


63° 45' 


38a5' 


3o84' 


9* 
79^ 
65* 


.897033 


3o83'.33- 


1918' 


18 


6r 3o / 


4050* 


3I7T 


.934084 


3i76 7 .07- 


3133' 


»9 


7i° i5> 


42 7 5' 


3356^ 


.947063 


3355'.3i- 


3333' 


30 


75* 


45oo' 


333i' 


5i' 


.965969 


&2&.61 


3548' 


31 


78* 45' 


47*5* 


33 7 2' 


37* 
33' 


.980803 


337 1 '.70 


3767' 


33 


83* 3C 


495o» 


3409* 


.991565 


34o8*.34- 


3989* 


33 


86* i5' 


5i75' 


343i' 


i 


•997964 


343C.39- 


3313* 


*4 


9°* 


5400* 


3438* 


1.000000 


3437'.75 


3438' 



The rule by which the sines are, in the text, directed to be found, may 
be illustrated as follows. Let «, a', s ", «'", *"" etc., represent the succes- 
sive sines. The first of the series, «, is assumed to be equal to its arc, or 
225', from which quantity, as is shown in the table above, it differs only 
by an amjnnt much smaller than the table takes any account of. Then 

«' = 1 +«-- 

$ $ 

« «' «" ft'" 

ft ft ft ft 



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198 K Burgess, etc., [ii.27. 

and so on, through the whole series, any fraction larger than a half being 
counted as one, and ^smaller fraction being rejected. In the majority of 
cases, as is made evident by the table, this process yields correct results : 
we have marked in the column of " true sines" with a plus or minus sign 
such modern values of the sines as differ by more than half a minute 
from those assigned by the Hindu table. 

It is not to be supposed, however, that the Hindu sines were originally 
obtained by the process described in the text. That process was, in all 
probability, suggested by observing the successive differences in the values 
of the sines as already determined by other methods. Nor is it difficult 
to discover what were those methods ; they are indicated by the limita- 
tion of the table to arcs differing from one another by 3° 45', and by 
what we know in general of the trigonometrical methods of the Hindus. 
The two main principles, by the aid of which the greater portion of all 
the Hindu calculations are made, are, on the one hand, the equality of the 
square of the hypo then use in a right-angled triangle to the sum of the 
squares of the other two sides, and, on the other hand, the proportional 
relation of the corresponding parts of similar triangles. The first of these 
principles gave the Hindus the sine of the complement of any arc of 
which the sine was already known, it being equal to the square root of 
the difference between the squares of radius and of the given sine. This 
led farther to the rule for finding the versed sine, which is given above in 
the text : it was plainly equal to the difference between the sine comple- 
ment and radius. Again, the comparison of similar triangles showed that 
the chord of an arc was a mean proportional between its versed sine and 
the diameter ; and this led to a method of finding the sine of half any 
arc of which the sine was known : it was equal to half the square root 
of the product of the diameter into the versed sine. That the Hindus 
had deduced this last rule does not directly appear from the text of this 
Siddhanta, nor from the commentary of Ranganatha, which is the one 
given by our manuscript and by the published edition ; but it is distinctly 
stated in the commentary which Davis had in his hands (As. Res. ii. 247) ; 
and it might be confidently assumed to be known upon the evidence of 
the table itself; for the principles and rules which we have here stated 
would give a table just such as the one here constructed. The sine 
of 90° was obviously equal to radius, and the sine of 30° to half radius : 
from the first could be found the sines of 45°, 22° 30', and 11° 15'; 
from the latter, those of 15°, 7° 30', and 8° 45'. The sines thus ob- 
tained would give those of the complementary arcs, or of 86° 15', 82° 
30', 78° 45', 75 Q , etc.; and the sine of 75°, again, would give those of 
37° 30' and 18° 45'. By continuing the same processes, the table of sines 
would soon be made complete for the twenty-four divisions ofrfhe quad- 
rant ; but these processes could yield nothing farther, unless by intro- 
ducing fractions of minutes; which was undesirable, because the symmetry 
of the table would thus be destroyed, and no corresponding advantage 
gained ; the table was already sufficiently extended to furnish, by inter- 
polation, the sines intermediate between those given, with all the accu- 
racy which the Hindu calculations required. 

If, now, an attempt were made to ascertain a law of progression for 
the series, and to devise an empirical rule by which its members might 



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ii.27.] 



S&rya-Siddhdnta. 



199 



be developed, the one from the other, in order, nothing could be more 
natural than to take the differences of the successive sines, and the differ- 
ences of those differences, as we have given them under the headings A' 
and A" in the annexed table. 

Hindu Sines, with their First and Second Differences. 



No. 


Sine. 


A' 


A" 


No. 


Sine. 


A' j A" 


o 

I 
a 
3 
4 
5 
6 

7 
8 

9 

IO 

ii 

12 


ooo 
aa5 
449 
671 
890 
no5 
i3i5 
i5ao 
1719 
1910 
3093 
a 267 
a43i 


aa5 
aa4 
aaa 
219 
ai5 
aio 
ao5 

199 
191 
i83 
174 
164 


1 
a 
3 
4 
5 
5 
6 
8 
8 

9 
10 
10 


ia 
i3 
i4 
i5 
16 
17 
18 

*9 
ao 
ai 
22 

23 

24 


243i 
2585 
2728 
2859 
2978 
3o84 
3i77 
3256 
3321 
33 7 2 
3409 
343i 
3438 


i54 

i43 

i3i 

119 

106 

93 

79 

65 

5i 

37 
aa 

7 


10 
11 
ia 
ia 
i3 
i3 
i4 
i4 
i4 
i4 
i5 
i5 



With these differences before him, an acute observer could hardly fail 
to notice the remarkable fact that the differences of the second order in- 
crease as the sines ; and that each, in fact, is about the ^j^th part of the 
corresponding sine. Now let the successive sines be represented by 0, «, 
*', *", *'", *"", and so on ; and let q equal jjy, or | ; let the first differ- 
ences be rf=«-0, (*'=*'-*, d"=8"-s' f (*"'=*'"-«", etc The sec- 
ond differences will be: -9q=d'-di-jq=d"-d\ -trqz=f"-d'\ 
etc. These last expressions give 

d' =rf — *q =« — *q 

J'"= d"- •"gas*- sq - *'q - $"q, etc 



Hence, also, 



•' =* -\-d' =1 + «-tg 

«" = t' + d" = «' + * - sq - s'q 

«'"= «"+ d'"= «"+ 8 - sq - «'g- 9 "q, 



and so on, according to the rule given in the text 

That the second differences in the values of the sines were proportional 
to the sines themselves, was probably known to the Hindus only by ob- 
servation. Had their trigonometry sufficed to demonstrate it, they might 
easily have constructed a much more complete and accurate table of 
sines. We add the demonstration given by Delambre (Histoire de 1' As- 
tronomic Ancienne, i. 458), from whom the views here expressed have 
been substantially taken. 

Let a be any arc in the series, and put 3° 45' = n. Then sin (a— »), 
sin a, sin (a + n), will be three successive terms in the series: sin 
a— sin (a— n), and sin (a + n) — sin a, will be differences of the first 
order ; and their difference, sin (a + n) -f- sin (a — n)— 2 sin a, will 
be a difference of the second order. But this last expression, by virtue 



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200 R Burgess, etc., [il 27. 

of the formula 22sin(ad=n) = sin a cosn =fc oosasinn, reduces to 

2 sin a cos n-r-fi— 2 sin a, or 2i -^- — 11 sin a. That is to say, tLe 

second difference is e<jual to the product of the sine of the arc a into a 
certain constant quantity, or it vanes as the sine. When n equals 3° 45', 
as in the Hindu table, it is easy to show, upon working out the last ex- 
pression by means of the tables, that the constant factor is, as stated by 
Delambre, a3 V & , instead of being ^J^, as empirically determined by 
the Hindus. 

It deserves to be noticed, that the commentary of Rangan&tha recog- 
nizes the dependence of the rule given in the text upon the value of the 
second differences. According to him, however, it is by describing a 
circle upon the ground, laying off the arcs, drawing the sines, and deter- 
mining their relations by inspection, that the method is obtained. The 
differences of the sines, he says, will be observed to decrease, while the 
differences of those differences increase ; and it will be noticed that the 
last second difference is 15' 16" 48'". A proportion is then made : if at 
the radius the second difference is of this value* what will it be at any 
sine? or, taking the first sine as an example, 3438' ; 15' 16" 48'" : : 225 
: 1. Nothing can be clearer, however, than that this pretended result of 
inspection is one of calculation merely. It would be utterly impossible 
to estimate by the eye the value of a difference with such accuracy, and, 
were it possible, that difference would be found very considerably removed 
from the one here given, being actually only about 14' 45". The value 
15' 16" 48'" is assumed only in order to make its ratio to the radius 

ewwtiy sir- 

The earliest substitution of the sines, in calculation, for the chords, 
which were employed by the Greeks, is generally attributed (see Whewell's 
History of the Inductive Sciences, B. ILL ch. iv. 8) to the Arab astron- 
omer Albategnius (al-Batt&nl), who flourished in the latter part of the 
ninth century of our era. It can hardly admit of question, however, 
that sines had already at that time been long employed by the Hindus. 
And considering the derivation by the Arabs from India of their system 
of notation, and of so many of the elements of their mathematical 
science, it would seem not unlikely that the first hint of this so conveni- 
ent and practical improvement of the methods of calculation may also 
have come to them from that country. This cannot be asserted, however, 
with much confidence, because the substitution of the sines for the chords 
seems so natural and easy, that it may well enough have been hit upon 
independently by the Arabs ; it is a matter for astonishment, as remarked 
by Delambre (Histoire de FAstronomie du Moyen Age, p. 12}, that 
Ptolemy himself who came so near it, should have failed of it If 
Albategnius got the suggestion from India, he, at any rate, got no more 
than that His table of sines, much more complete than that of the 
Hindus, was made from Ptolemy's table of chords, by simply halving them. 
The method, too, which in India remained comparatively barren, led to 
valuable developments in the hands of the Arab mathematicians, who 
went on by degrees to form also tables of tangents and co-tangents, secants 
and co-secants ; while the Hindus do not seem to have distinctly appreci- 
ated the significance even of the cosine. 



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ii. 28.] S&rya-Siddh&nta. 201 

In this passage, the sine is called jy&rdha, "half-chord;" hereafter, 
however, that term does not once occur, but^yd u chord " (literally " bow- 
string") is itself employed, as are also its synonyms jiv&, m&urvikd, to 
denote the sine. The usage of Albategnius is the same. The sines of the 
table are called pinda, or jy&pinda, "the quantity corresponding to the 
sine." The term used for versed sine, utkramajyd, means " inverse-order 
sine," the column of versed sines being found by subtracting that of 
tines in inverse order from radius. 

The ratio of the diameter to the circumference involved in the expres- 
sion of the value of radius by 3438' is, as remarked above (under i. 59, 
60), 1 : 3.14186. The commentator asserts that value to come from the 
ratio 1250 : 3927, or 1 : 3.1416, and it is, in fact, the nearest whole num- 
ber to the result given by that ratio. If the ratio were adopted which 
has been stated above (in i. 59), of 1 : J 10, the value of radius would be 
only 3415'. It is to be observed with regard to this latter ratio, that it 
could not possibly be the direct result of any actual process adopted for 
ascertaining the value of the diameter from that of the circumference, or 
the contrary. It was probably fixed upon by the Hindus because it 
looked and sounded well, and was at the same time a sufficiently near 
approximation to the truth to be applied in cases where exactness was 
neither attainable by their methods, nor of much practical consequence ; 
as in fixing the dimensions of the earth, and of the planetary orbits. 
The nature of the system of notation of the Hindus, and their constantly 
recurring extraction of square roots in their trigonometrical processes, 
would cause the suggestion to them, much more naturally than to .the 
Greeks, of this artificial ratio, as not far from the truth ; and their science 
was just of that character to choose for some uses a relation expressed in 
a manner so simple, and of an aspect so systematical, even though known 
to be inaccurate. We do not regard the ratio in question, although so 
generally adopted among the Hindu astronomers, as having any higher 
value and significance than this. 

28. The sine of greatest declination is thirteen hundred and 
ninety-seven ; by this multiply anjr sine, and divide by radius ; 
the arc corresponding to the result is said to be the declination. 

The greatest declination, that is to say, the inclination of the plane of 
the ecliptic, is here stated to be 24°, 1397' being the sine of that angle. 
The true inclination in the year 300 of our era, whicli we may assume 
to have been not far from the time when the Hindu astronomy was 
established, was a little less than 23° 40', so that the error of the Hindu 
determination was then more than 20' : at present, it is 32' 34". The 
value assigned by Ptolemy (Syntaxis, i) to the inclination was between 
23° 50* and 23° 52' 30" ; an error, as compared with its true value in 
the time of Hipparchus, of only about V. 

The second naif of the verse gives, in the usual vague and elliptical 
language of the treatise, the rule for finding the declination of any given 
point in the ecliptic. We have not in this case supplied the ellipses in 
our translation, because it could not be done succinctly, or without 
introducing an element, that of the precession, which possibly was not 
taken into account when the rule was made. See what is said upon this 
vol. vi. 96 



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202 



E. Burgess, etc,. 



[ii. 28- 



subject under verses 9 and 10 of the next chapter. The "sine" em- 
ployed is, of course, the sine of the distance from the vernal equinox, or 
of the longitude as corrected by the precession. 

The annexed figure will explain the rule, and the method of its 
demonstration. 

Let ACE represent a quadrant of the plane of the equatorial, and 
ACG a quadrant of that of the ecliptic, AC being the line of their 
intersection : then A P is the equinoctial colure, P E the solstitial, G E, 
or the angle G C E, the inclination of the ecliptic, or the greatest decli- 
nation (param&pakrama, or paramakrdnti), and GD its sine (parama- 
kr&ntjjyi). Let 8 be the position of the sun, and draw the circle of 
declination P H ; &H, or the angle 
S C H, is the declination of the sun 
at that point* and SF the sine of I 
declination (kr&ntijyd). From 8 and - 
F draw 8 B and F B at right angles I 
to A C ; then 8 B is the sine of the | 
arc A 8, or of the sun's longitude. 
But GCD and 8BF are similar I 
right-angled triangles, having their 
angles at C and B each equal to the | 
inclination. Therefore C G : G D : : 

SB:SF; and SF = ^1^; 

- A , . . , sin incl.X sin long. 

that is, sin decl. := - 2. . 

K 

The same result is, by our modern 

methods, obtained directly from the formula in right-angled spherical 

trigonometry: sine —sin a sin C; or, in the triangle ASH, right-angled 

at H, sin SH — sin S A sin S AH. 

29. Subtract the longitude of a planet from that of its apsis 
(mandocca) ; so also, subtract it from that of its conjunction 
(cighra); the remainder is its anomaly (kendra); from that is 
found the quadrant (pada) ; from this, the base-sine (bhujajyd), 
and likewise that of the perpendicular (Jcoti). 

80. In an odd (vishama) quadrant, the base-sine is taken from 
the part past, the perpendicular from that to come ; but in an 
even (yugma) quadrant, the base-sine (bdhujyd) is taken from 
the part to come, and the perpendicular-sine from that past 

The distance of a planet from either of its two apices of motion, or 
centres of disturbance, is called its kendra ; according to the comment* 
ary, its distance from the apsis (mandocca) is called mandakendra, and 
that from the conjunction (ptghrocca) is called fighrakendra : the Surya- 
Siddhanta, however, nowhere has occasion to employ these terms. The 
former of the two corresponds to what in modern astronomy is called 
the anomaly, the latter to what is known as the commutation. The 
word kendra is not of Sanskrit origin, but is the Greek xirTgor ; it is a 
circumstance no less significant to meet with a Greek word thus at the 




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ii. 30.] 



Suinja-Siddh&nla. 



203 



very foundation of the method of calculating the true place of a planet 
by means of a system of epicycles, than to find one, as noticed above 
(under i. 52), at the base of the theory of planetary regency upou which 
depend the names and succession of the days of the week. Both 
anomaly and commutation, it will be noticed,, are, according to this 
treatise, to be reckoned always forward from the planet to its apsis and 
conjunction respectively ; excepting that, in the case of Mercury and 
Venus, owing to the exchange with regard to those planets of the place 
of the planet itself with that of its conjunction, the commutation is 
really reckoned the other way. The functions of any arc being the 
same with those of its negative, it makes no difference, of course, 
whether the distance is measured from the planet to the apex {wxa) y or 
from the apex to the planet. 

Hie quantities actually made use of in the calculations which are to 
follow are the sine and cosine of the anomaly, or of the commutation. 
The terms employed in the text require a little explanation. Bhuja 
means " arm ;" it is constantly applied, as are its synonyms bdhu and 
dos, to designate the base of a right-angled triangle ; koli is properly 
" a recurved extremity," and, as used to signify the perpendicular in 
such a triangle, is conceived of as being the end of the bhvja y or base, 
bent op to an upright position : bhujajyd and kotijyd, then, are literally 
the values, as sines, of the base and perpendicular of a. right-angled 
triangle of which the hypothenuse is made radius : owing to the relation 
to one another of the oblique angles of such a triangle, they are re- 
spectively as sine and cosine. We have not been willing to employ 
these latter terms in translating them, because, as before remarked, the 
Hindus do not seem to have conceived of the cosine, the sine of the 
complement, of an arc, as being a function of the arc itself. 

To find the sine and cosine of the planet's distance from either of its 
apices (ticca) is accordingly the yj g 2 . 

object of the directions given in 
verse SO and the latter part of 
the preceding verse. The rule 
itself is only the awkward Hindu 
method of stating the familiar 
truth that the sine and cosine of 
an arc and of its supplement are 
equal. The accompanying figure 
will, it is believed, illustrate the 
Hindu manner of looking at the 
subject Let P be the place of a 
planet, and divide its orbit into 
the four quadrants P Q, Q R, R S, 
and S P ; the first and third of 
these are called the odd (vishama) 
quadrants; the second and fourth, 
tne even (yugma) quadrants. Let A, B, C, and D, be four positions of 
the apsis (or of the conjunction); then the arcs PA, PQB, PQRC, 
PQRSD will be the values of the anomaly in each case. AM, the 
base-sine, or sine of anomaly, when the apsis is in the first quadTant, is 




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204 K Burgess, etc., [ii. 30- 

detennined by the arc A P, the arc passed over in reckoning the anom- 
aly, while AG or E M, the perpendicular-sine, or cosine, is taken from 
the arc A Q, the remaining part of the quadrant The same is true in 
the other odd quadrant, R S; the sine C H, or E L, comes from R C, 
the part of the quadrant between the planet and the apsis; the cosine 
C L is from its complement. But in the even quadrants, Q R and S P, 
the case is reversed ; the sines, BH, or EF, and DM, are determined by 
the arcs B R and £> P, the parts of the quadrant not included in the 
anomaly, and the cosines, B F and E D, or E M, correspond to the other 
portions of each quadrant respectively. 

This process, of finding what portion of any arc greater than a quad- 
rant is to be employed m determining its sine, is ordinarily called in 
Ilindu calculations " taking the bhuja of an arc." 

31. Divide the minutes contained in any arc by two hundred 
and twenty-five ; the quotient is the number of the preceding 
tabular sine (jy&piTidaKa). Multiply the remainder by the differ- 
ence of the preceding and following tabular sines, and divide 
by two hundred and twenty-five ; 

32. The quotient thus obtained add to the tabular sine called 
the preceding ; the result is the required sine. The same method 
is prescribed also with respect to the versed sines. 

33. Subtract from any given sine the next less tabular sine; 
multiply the remainder by two hundred and twenty-five, and 
divide by the difference between the next less and next greater 
tabular sines ; add the quotient to the product of the serial num- 
ber of the next less sine into two hundred and twenty-five: the 
result is the required arc. 

The table of sines and versed sines gives only those belonging to arcs 
which are multiples of 3° 45' ; the first two verses of this passage state 
the method of finding, by simple interpolation, the sine or versed sine 
of any intermediate arc ; while the third verse gives the rule for the 
contrary process, for converting any given sine or versed sine in the 
same manner into the corresponding arc. 

In illustration of the first rule, let us ascertain the sine corresponding 
to an arc of 24°, or 1440 / . Upon dividing the latter number by 225, 
we obtain the quotient 6, and the remainder 90*. This preliminary step 
is necessary, because the Hindu table is not regarded as containing any 
designation of the arcs to which the sines belong, but as composed 
simply of the sines themselves in their order. The Bine corresponding 
to the quotient obtained, or the sixth, is 1315' : the difference between 
it and the next following sine is 205'. Now a proportion is made : it, 
at this point in the quadrant, an addition of 225' to the arc causes an 
increase in the sine of 205', what increase will be caused by an addition 
to the arc of 90' : that is to say, 225 : 205 : : 90 : 82. Upon adding the 
result, 82', to the sixth sine, the amount, 1397', is the sine of the given 
arc, as stated in verse 28. The actual value, it may be remarked, of 
the sine of 24°, is 1398'.26. 

The other rule is the reverse of this, and does not require illustration. 



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it. 38.] Stlrya-Siddlidnta. 205 

The extreme conciseness aimed at in the phraseology of the text, and 
not unfrequently carried by it beyond the limit of distinctness, or even 
of intelligibility, is well illustrated by verse 33, which, literally trans- 
lated, reads thus : " having subtracted the sine, the remainder, multi- 
plied by 225, divided by its difference, having added to the product of 
the number and 225, it is called the arc. \n verse 31, also, the 
important word "remainder" is not found in the text 

The proper place for this passage would seem to be immediately after 
the table of sines and versed sines : it is not easy to see why verses 
28-30 should have been inserted between, or indeed, why the subject of 
the inclination of the ecliptic is introduced at all in this part of the 
chapter, as no use is made of it for a long time to come. 

34. The degrees of the sun's epicycle of the apsis {manda- 
paridht) are fourteen, of that of the moon, thirty -two, at the end 
of the even quadrants ; and at the end of the odd quadrants, 
they are twenty minutes less for both. 

35. At the end of the even quadrants, they are seventy-five, 
thirty, thirty-three, twelve, forty-nine ; at the odd (afa) they are 
seventy-two, twenty-eight, thirty-two, eleven, forty-eight, 

86. For Mars and the rest ; farther, the degrees of the epi- 
cycle of the conjunction (dghrd) are, at the end of the even 
quadrants, two hundred and thirty-five, one hundred and thirty- 
three, seventy, two hundred and sixty-two, thirty-nine ; 

87. At the end of the odd quadrants, they are stated to be 
two hundred and thirty-two, one hundred and thirty-two, 
seventy-two, two hundred and sixty, and forty, as made use of 
in the calculation for the conjunction (fyhraJcarrnan). 

38. Multiply the base-sine (bhujajyd) by the difference of the 
epicycles at tne odd and even quadrants, and divide by radius 
(trijy$)\ th© result, applied to the even epicycle (yrtta\ and 
additive (dhana) or subtractive (rna), according as this is less or 
greater than the odd, gives the corrected (epkuta) epicycle. 

The corrections of the mean longitudes of the planets for the dis- 
turbing effect of the apsis (mandocca) and conjunction (fighrocca) of 
each — that is to say, for the effect of the ellipticity of their orbits, and 
for that of the annual parallax, or of the motion of the earth in its 
orbit — are made in Hindu astronomy by the Ptolemaic method of epi- 
cycles, or secondary circles, upon the circumference of which the planet 
is regarded as moving, while the centre of the epicycle revolves about 
the general centre of motion. The details of the method, as applied by 
the Hindus, will be made clear by the figures and processes to be pre- 
sented a little later ; in this passage we have only the dimensions of the 
epicycles assumed for each planet. For convenience of calculation, they 
are measured in degrees of the orbits of the planets to which they 
severally belong; hence only their relative dimensions, as compared 
with the orbits, are given us. The data of the text belong to the planets 
in the order in which these succeed one another as regents of the days 



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206 



E. Burgess, etc., 



[ii. 38- 



of the week, viz., Mara, Mercury, Jupiter, Venus, and Saturn (see 
above, under i. £1, 52). The annexed table gives the dimensions of 
the epicycles, both their circumferences, which are presented directly 
by the text, and their radii, which we have calculated after the method 
of this Siddhanta, assuming the radius of the orbit to be 3438'. 

Dimensions of the Epicycles of the Planets. 





Epicycle of the apafc : 


Eoioyob of tbo conjunction : 


Planet. 


at art o quadrant, 


at odd qoadiant, 


at oven quadrant, 


at odd qoadrant, 




circ. 


rad. 


circ 


rad. 


circ 


rad. 


aire. 


rad. 


Sun, 


14° 


i3y.70 


i3°4o' 


i3o\52 


• . . . 


••••••• 


.... 


•*•••«. 


Moon, 


3a° 


3oy.6o 


3i Q 4o* 


3oaMa 


. • • . 





.... 





Mercury, 


3o* 


28ff.5o 


28° 


26Y.40 


i33° 


1270'. 1 5 


132° 


126V.60 


Venue, 


12* 


ii4'-6o 


11° 


io^-oS 


262° 


aSoa'.io 


260° 


2483'.oo 


Kara, 


75° 


7iff.a5 


7* Q 


68r6o 


235° 


2244'-25 


232° 


22l5'.6o 


Jupiter, 


33" 


scy.is 


3a* 


3o5'.6o 


70° 


668' 5o 


72° 


687.60 


Saturn, 


49° 


467'-95 


48* 


458^^0 


39° 


372^5 


4o° 


38a'.oo 



A remarkable peculiarity of the Hindu system is that the epicycles 
are supposed to contract their dimensions as they leave the apsis or the 
conjunction respectively (excepting in the case of the epicycles of the 
conjunction of Jupiter and Saturn, which expand instead of contracting), 
becoming smallest at the quadrature, then again expanding till the lower 
apsis, or opposition, is reached, and decreasing and increasing in like 
manner in the other half of the orbit ; the rate of increase and diminu- 
tion being as the sine of the distance from the apsis, or conjunction. 
Hence the rule in verse 38, for finding the true dimensions of the epi- 
cycle at any point in the orbit It is founded upon the simple propor- 
tion : as radius, the sine of the distance at which the diminution (or 
increase) is greatest, is to the amount of diminution (or of increase) at 
that point, so is the sine of the given distance to the corresponding 
diminution (or increase) ; the application of the correction thus obtained 
to the dimensions of the epicycle at the apsis, or conjunction, gives the 
true epicycle. 

We shall revert farther on to the subject of this change in the dimen- 
sions of the epicycle. 

The term employed to denote the epicycle, paridhi, means simply 
" circumference," or " circle ; " it is the same which is used elsewhere in 
this treatise for the circumference of the earth, etc. In a single instance, 
in verse 38, we have vrtia instead of paridhi ; its signification is the 
same, and its other uses are closely analogous to those of the more 
usual term. 

39. By the corrected epicycle multiply the base-sine (bhujajyd) 
and perpendicular-sine (jcotijyfl) respectively, and divide by the 
number of degrees in a circle : then, the arc corresponding to 
the result from the base-sine (bhuj'ajydphata) is the equation of 
the apsis (mdnda phdla), in minutes, etc 

All the preliminary operations having been already performed, this is 
the final process by which is ascertained the equation of the apsis, or 
the amount by which a planet is, at any point in its revolution, drawn 



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ii. 39.] S&rya-Siddhdnta. 207 

away from its mean place bv the disturbing influence of the apsis. In 
modern phraseology, it is called the first inequality, due to the ellipticity 
of the orbit ; or, the equation of the centre. 

Figure 3, upon the next page, will serve to illustrate the method of 
the process. 

Let A MM' P represent a part of the orbit of any planet, which is 
supposed to be a true circle, having E, the earth, for its centre. Along 
this orbit the planet would move, in the direction indicated by the 
arrow, from A through M and M' to P, and so on, with an equable 
motion, were it not for the attraction of the beings situated at the apsis 
(mandocca) and conjunction (ftphrocca) respectively. The general mode 
of action of these beings has been explained above, under verses 1-5 
of this chapter : we have now to ascertain the amount of the disturb- 
ance produced by them at any given point in the planet's revolution. 
The method devised is that of an epicycle, upon the circumference of 
which the planet revolves with an equable motion, while the centre of the 
epicycle traverses the orbit with a velocity equal to that of the planet's 
mean motion, having always a position coincident with the mean place 
of the planet At present, we nave to do only with the epicycle which 
represents the disturbing effect of the apsis (mandooca). The period of 
the planet's revolution about the centre of the epicycle is the time 
which it takes the latter to make the circuit of the orbit from the apsis 
around to the apsis again, or the period of its anomalistic revolution. 
This is almost precisely equal to the period of sidereal revolution in the 
case of all the planets excepting the moon, since their apsides are re- 
garded by the Hindus as stationary (see above, under i. 41-44) : the 
moon's apsis, however, has a forward motion of more than 40° in a 
year; hence the moon's anomalistic revolution is very perceptibly 
longer than its sidereal, being 27 d 18 k 18 m . The arc of the epicycle 
traversed by the planet at any mean point in its revolution is accord- 
ingly always equal to the arc of the orbit intercepted between that 
point and the apsis, or to the mean anomaly, when the latter is reckoned, 
in the usual manner, from the apsis forward to the planet. Thus, in the 
figure, suppose A to be the place of the apsis (mandooca, the apogee of 
the sun and moon, the aphelion of the other planets), and P that of the 
opposite point (perigee, or perihelion ; it has in this treatise no distinct- 
ive name); and let M and M' be two mean positions of the planet, or 
actual positions of the centre of the epicycle ; the lesser circles drawn 
about these four points represent the epicycle : this is made, in the figure, 
of twice the size of that assumed for the moon, or a little smaller than 
that of Mars. Then, when the centre of the epicycle is at A, the 
planet's place in the epicycle is at a ; as the centre advances to M, M', 
and P, the planet moves in the opposite direction, to m, *»', and jp, the 
arc a' m beinjj equal to A M, a" m' to A M', and a'"p to A P. It is as if, 
while the axis £a revolves about E, the part of it A a remained con- 
stant in direction, parallel to £ A, assuming the positions M m, M' m f , 
and Vp successively. The effect of this combination of motions is to 
make the planet virtually traverse the orbit indicated in the figure by 
the broken line, which is a circle of equal radius with the true orbit, 
but having its centre removed from E, toward A, by a distance equal to 



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208 



M Burgess, etc., 
Ffc. 3. 



[ii, 30. 



_. D _ 1 




^^ 


* 




^V A 




ft 


v* s y^ / ^ 




r 


- P?K 














■\ 




4 




\ 


/ 


•• 


X ■■ 


H 




i •• 


V •' 



A «, the radius of the epicycle. This identity of the virtual orbit with 
an eccentric circle, of which the eccentricity is equal to the radius of 
the epicycle, was doubtless known to the Hindus, as to Ptolemy : the 
latter, in the third book of his Syntaxis, demonstrates the equivalence of 
the suppositions of an epicycle and an eccentric, and chooses the latter 
to represent the first inequality : the Hindus have preferred the other 
supposition, as better suited to their methods of calculation, and as ad- 
mitting a general similarity in the processes for the apsis and the con- 
junction. The Hindu theory, however, as remarked above (under w. 
1-5 of this chapter), rejects the idea of the actual motion of the planet 
in the epicycle, or on the eccentric circle : the method is but a device 
for ascertaining the effect of the attractive force of the being at the 
apsis. Thus the planet really moves in the circle AMM'P, and if the 
lines Em, E m' be drawn, meeting the orbit in o and o 7 , its actual place 
is at o and 7 , when its mean place is at M and M ; respectively. To 
ascertain the value of the arcs oM and I'M', which are the amount of 
removal from the mean place, or the equation, is the object of the pro- 
cess prescribed by the text. • ~ 
Suppose the planet's mean place to be M, its mean distance from the 
apsis being A M : it has traversed, as above explained, an equal arc, a'ti^ 
in the epicycle. From M draw M B and M F, and from m draw m n, 
at right angles to the lines upon which they respectively fall : then M B 
is the base-sine (bhujajyd), or the sine of mean anomaly, and MF, or its 
eaual E B, is the perpendicular-sine (kotjjyd), or cosine, and m n and 
n M are corresponding sine and cosine in the epicyele. But as the rela- 
tion of the circumference of the orbit to that of the epicycle is known, 
and as all corresponding parts of two circles are to one another as their 
respective circumferences, the values of m n and n M are found by a 
proportion, as follows : as 860° is to the number of degrees in the cir- 
cumference of the epicycle at M, so is MB town, and E B to n M. 
Hence mn is called the tt result from the base-sine" (bhujajyAphala^ or, 
more briefly, bhujaphala, or b&kupkala), and n M the " result from the 
perpendicular-sine" (kotijydphala y or kotiphala) : the latter of the two, 
however, is not employed in the process for calculating the equation of 
the apsis. Now, as the dimensions of the epicycle of the apsis are in 



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5 


35 3 7 


II 


ao 


59 


i 


IO 


9 


4a 


a6 






a 


5o 


IO 


9 


45 


16 


11 


ao 


59 


i 



ii.39.] Stirya-Siddhdnta. 209 

all cases small, m n may without any considerable error be assumed to 
be equal to o g, which is the sine of the arc o M, the equation : this 
assumption is accordingly made, and the conversion of m n, as sine, into 
its corresponding arc, gives the equation required. 

The same explanation applies to the position of the planet at M' : a" 
wi', the equivalent of A M M', is here the arc of the epicycle traversed ; 
m' *', its sine, is calculated from M' B', as before, and is assumed to 
equal & q', the sine of the equation & M'. 

To rive a farther and practical illustration of the process, we will 
proceed to calculate the equation of the apsis for the moon, at the time 
for which her mean place has been found in the notes to the last chap- 
ter, vi&, the 1st of January, 1860, midnight, at Washington. 

Moon's mean longitude, midnight, at Ujjayinl (L 63), n 8 i5° a3' iA" 

add the equation for difference of meridian (dtfdniarapkala\ ) 
or for her motion between midnight at Ujj. and Wash. (i. 60, 61), ) 

Moon's mean longitude at required time, 

Longitude of moon's apsis, midnight, at Ujjayinl (L 68), 
add for difference of meridian, as above, 

Longitude of moon's apeis at required time, 
deduct moon's mean longitude (ii. 29), 

Moon's .mean anomaly (mandak*ndra\ io 18 46 i5 

The anomaly being reckoned forward on the orbit from the planet, 
the position thus found for the moon relative to the apsis is, nearly 
enough for purposes of illustration, represented by M in the figure. By 
the rule given above, in verse 30, the base-sine (bkujajyd) — since the 
anomaly is in the fourth, an even, quadrant — is to be taken from the 
part of the quadrant not included in the anomaly, or AM; the per- 
pendicular-sine (kotijyd) is that corresponding to its complement, or 
MD. That is to say: 

From the anomaly, • i©« t8° 48' i$" 

oeduct three quadrants, o 

remains the arc M D, i 18 46 i5 

take this from a quadrant, 3 

remains the arc A M, i n i3 49 

And by the method already illustrated under verses 81, 32, the sine 
corresponding to the latter arc, which is the base-sine (bkujajyd), or the 
sine of mean anomaly, M B, is found to be 2266' ; that from M D, which 
is MF, or EB, the perpendicular-sine (kotijyd), or cosine of mean 
anomaly, is 2585'. 

The next point is to find the true size of the epicycle at M. By 
verse 34, the contraction of its circumference amounts at D to 20'; 
hence, according to the rule in verse 38, we make the proportion, sin 
A D : 20' : : sin A M : diminution at M ; or, 

3438:ao::aa66:i3 
Deducting from 32°, the circumference of the epicycle at A, the amount 
of diminution thus ascertained, we have 31° 47' as its dimensions at M. 
vol. vi. 27 



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210 E. Burgess, etc., [ii. 39- 

Once more, by verse 39, we make the proportion, circ. of orbit.; circ. 
of epicycle : : M B : m n ; or, 

36o° : 3i° 47' : : 2260" ; 200 

The value, then, of mn, the result from the base-sine (bhujajy&phala), 
is 200' ; which, as m n is assumed to equal o g, is the sine of the equa- 
tion. Being less than 225', its arc (see the table of sines, above) is of 
the same value : 3° 20', accordingly, is the moon's equation of the apsis 

fmdnda phala) at the given time : the figure shows it to be subtractive 
rna), as the rule in verse 45 also declares it. Hence, from the 

Hood's mean longitude, n* 20° 59' 

deduct the equation, 3 20 

Moon's true longitude, 11 17 39 

We present below, in a briefer form, the results of a similar calcula- 
tion made for the sun, at the same time. 

Sun's mean longitude, midnight, at Ujjayini (I 58), 8« 17° 48' 7" 

add for difference of meridian (i 60, 61), i5 6 

Sun's mean longitude at required time, 8 18 i3 i3 

Longitude of sun's apsis (L 41), 2 17 17 34 

Sun's mean anomaly (ii. 29), 5 29 4 n 

subtract from two quadrants (ii. 80), 6 

Arc determining base-sine, 55' 49" 

Base-sine (bkujajyd), 56' 

Dimensions of epicycle (ii. 88), tf* 
Result from base-sine (bhujajydphala), or sine of equation (ii. 89), a' 

Equation (mdnda phala, ii. 46), +2' 

Sun's true longitude, 8 s 18 9 i5' 

In making these calculations, we have neglected the seconds, rejecting 
the fraction of a minute, or counting it as a minute, according as it was 
less or greater than a half. For, considering that this method is followed 
in the table of sines, which lies at the foundation of the whole process, 
and considering that the sine of the arc in the epicycle is assumed to be 
equal to that of the equation, it would evidently be a waste of labor, and 
an affectation of an exactness greater than the process contemplates, or 
than its general method renders practicable, to carry into seconds the data 
employed. 

As stated below, in verse 43, the equation thus found is the only one 
required in determining the true longitude of the sun and of the moon : 
in the case of the other planets, however, of which the apparent place is 
affected by the motion of the earth, a much longer and more complicated 
process is necessary, of which the explanation commences with the next 
following passage. 

The Ptolemaic method of making the calculation of the equation of 
the centre for the sun and moon is illustrated by the annexed figure 
(Fig. 4). The points E, A, M, a, m, and 0, correspond with those simi- 
larly marked in the last figure (Fig. 3). The centre of the eccentric 



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ii. 42.] S&rya-Siddh&nta. 211 

circle is at e, and E*, which equals A a, is the eccentricity, which is given. 
Join *m; the angle mea equals ME A, the mean anomaly, and Em* 
equals MEo, the equation. Extend 

melody where it meets E d , a per- Fig. 4. 

pendieular let fall upon it from E. 
Then, in the right-angled triangle 
E«rf, the side Ee and the angles 
— since Eid equals mea — are 
given, to find the other sides, id 
and dE. Add t d to e m, the ra- 
dius ; add the square of the sum 
to that of Erf; the square root 
of their sum is E m : then, in the 
right-angled triangle mEd y all 
the sides and the right angle are 
given, to find the angle Erne, the 
equation. 

This process is equivalent to a transfer of the epicycle from M to E ; 
Ed becomes the result from the base-sine (bhujajy&phala), and de that 
from the perpendicular-sine (kotijyaphala), and the angle of the equation 
is found in the same manner as its sine, e c, is found in the Hindu process 
next to be explained ; while, in that which we have been considering, Ed 
is assumed to be equal to e c. 

Ptolemy also adds to the moon's orbit an epicycle, to account for her 
second inequality, the evection, the discovery of which does him so much 
honor. Of this inequality the Hindus take no notice. 

40. The result from the perpendicular-sine (Jcotjphala) of the 
distance from the conjunction is to be added to radius, when the 
distance (kcndra) is in the half-orbit beginning with Capricorn ; 
but when in that beginning with Cancer, the result from the 
perpendicular-sine is to be subtracted. 

41. To the square of this sum or difference add the square of 
the result from the base-sine (bdhuphala) ; the square root of 
their sum is the hypothenuse Qcarna) called variable (cala). 
Multiply the result from the base-sine by radius, and divide by 
the variable hypothenuse : 

42. The arc corresponding to the quotient is, in minutes, etc., 
the equation of the conjunction (gdighrya phah) ; it is employed 

n the first and in the fourth process of correction (karman) for 
Mars and the other planets. 

The process prescribed by this passage is essentially the same with that 
explained and illustrated under the preceding verse, the only difference 
being that here the sine of the required equation, instead of being 
assumed equal to that of the arc traversed by the planet in the epicycle, 
is obtained by calculation from it. The annexed figure (Fig. 5) will ex- 
hibit the method pursued. 

The larger circle, CMM'O, represents, as before, the orbit in which 
any one of the planets, as also the being at its conjunction (ptghrocca) are 



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212 



E. Burgess, etc., 
Ffc.c 



pi. 42. 




making the circuit of the heavens about E, the earth, as a centre, in the 
direction indicated by the arrow, from C through M and M' to O, and so 
on. But sinee, in every case, the conjunction moves more rapidly east- 
ward than the planet, overtaking and passing it, if we suppose the con- 
junction stationary at 0, the virtual motion of the planet relative to that 
point is backward, or from O through M' and M to C, its mean rate of 
approach toward C being the difference between the mean motion of the 
planet and that of the sun. As before, the amount to which the planet 
is drawn away from its mean place toward the conjunction ib calculated 
by means of an epicycle. The circles drawn in the figure to represent 
the epicycle are of the relative dimensions of that assigned to Mercury, 
or a little more than half that of Mars. The direction of the planet's 
motion in the epicycle is the reverse of that in the epicycle of the apsis, 
as regards the actual motion of the planet in its orbit, being eastward at 
the conjunction ; as regards the motion of the planet relative to the con- 
junction, it is the same as in the former case, being in the contrary direc- 
tion at die conjunction : its effect, of course, is to increase the rate of the 
eastward movement at that point The time of the planet's revolution 
about the centre of the epicycle is the interval between two successive 
passages through the point C, the conjunction : that is to say, it is equal 
to the period of synodical revolution of each planet. These periods are, 
according to the elements presented in the text of this Siddhanta, aa 
follows : 

Mercury, 

Venus, 

Mars, 

Jupiter, 

Saturn, 

The arc of the epicycle traversed by the planet, at any point in its revo- 
'taon, is equal to its distance from the conjunction, when reckoned for- 
-4- from the planet, according to the method prescribed in verse 29. 



u5* 


aik 


4*» 


583 


21 


37 


779 


22 


11 


398 


21 


20 


378 


2 


4 



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ii^42 A ] Stirya-Siddli&tita. 218 

Suppose, now, the mean place of -the planet, relative to its conjunction 
(ptgkrocca) at C, to be at M : its place in the epicycle is at m, as far from 
c'", in eitner direction, as M from C. The arc of the epicycle already 
traversed is indicated in this figure, as in Fig. 3, by the heavier line. 
Draw £ m, cutting the orbit in o ; then o is the planet's true place, and 
o M is the equation, or the amount of removal from the mean place by 
the attraction of the being at C. 

The sine and cosine of the distance from the conjunction, the dimen- 
sions of the epicycle, and the value of the correspondents in the epicycle 
to the sine and cosine, are found as in the preceding process. Add nlf, 
the result from the cosine (koffiyApkala), to M E, the radius : the result 
is the perpendicular, En, of the triangle Enw. To the square of En 
add that of the base n m, the result from the sine (bhujajy&pkala) ; the 
square root of the sum is the line E m, the hypo then use : it is termed the 
variable hypothenuse (cola karna) from its constantly changing its 
length. We have now the two similar triangles Emn and Eoy, a 
comparison of the corresponding parts of which gives us the proportion 
Em:mn::Eo:og; that is to say, o g y which is the sine of the equation 
oM, equals* the product of Eo, the radius, into mn, the result from the 
ba»e-sine* divided -by the variable hypothenuse, Em. 

-When the planet 1 * mean place is in the quadrant D O, as at M', the 
result from the perpendicular-sine (kotijydphala), or M' n', is subtracted 
from radius, and the remainder, E n\ is employed as before to find the 
value of Em', the variable hypothenuse: and the comparison of the 
similar triangles EmV and Eo'g' gives o' g\ the sine of the equation, 
o'W. 

It is obvious that when the mean distance of a planet from its conjunc- 
tion is less than a quadrant in either direction, as at M, the base En is 
greater than radius ; when that distance is more than a quadrant, as at 
M', the base En' is less than radius: the cosine is to be added to radius 
in- the one case, aud subtracted from it in the other. This is the mean- 
ing of the rule in verse 40 : compare the notes to i. 58 and ii. SO. 

In illustration of the process, we will calculate the equation of the 
conjunction of Mercury for the given time, or for midnight preceding 
January 1st, 1860, at Washington. 

Since the Hindu system, like the Greek, interchanges in the case of the 
two inferior planets the motion and place of the planet itself and of the 
sun, giving to the former as its mean motion that which is the mean 
apparent motion of the sun, and assigning to the conjunction (fighrocea) 
a revolution which is actually that of the planet in its orbit, the mean 
position of Mercury at the given time is that found above (under v. 80) 
to be that of thu sun at the same time, while to find that of its conjunc- 
tion we have to add the equation for difference of meridian (def&ntara- 
phala, i. 60, 61), to the longitude given under i. 58 as that of the planet. 

Longitude of Mercury's conjunction (fighrocea), midnight, at TJjjayini, 4* i5° i3' 8" 
add for difference of meridian, i 44 i4 



Longitude of conjunction at required time, 4 16 57 22 

Mean longitude of Mercury, 8 18 i3 i3 

Mean commutation (ftyArdtotdra,), 7 28 44 9 



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214 



E. Burgess, etc.. 



[ii.42- 



The position of Mercury with reference to the conjunction is accord- 
ingly very nearly that of M\ in Fig. 5. The arc which determines the 
hase-sine (bkujajyd), or O M', is 58° 44', while M' D, its complement, 
from which the perpendicular-sine (kotijyti) is taken, is 31° 16'. The 
corresponding sines, M' B' and M' G, are 2938' and 1784' respectively. 

The epicycle of Mercury is one degree less at D than at O. Hence 
the proportion 

3438 :6o:: 2^38 :5i 

gives 51' as the diminution at M' : the circumference of the epicyle at M, 
then, is 132° 9'. The two proportions 

36o°:i3a° 9':: 2938:1078, and 36o°.: i32°9':: 1784: 655, 
give us the value of m'n' as 1078', and that of n'M' as 655'. The 
commutation being more than three and less than nine signs, or in the 
half-orbit beginning with Cancer, the fourth sign, n' M' is to be sub- 
tracted from EM', or radius, 3438' ; the remainder, 2783', is the perpen- 
dicular E n'. 



To the square of En', 
add the square of n'm', 



add the square < 
of their ram, 



7,745,089 
1,162,084 

8,907,173 



the square root, 2984 

is the variable hypothenuse (cafa karna), E m'. The comparison of the 
triangles E m' »' and E o'g' gives the proportion E m* : m f n' : : E & : o'g 1 , or 

2984: 1078:: 3438: 1242 
The value of o'g', the sine of the equation, is accordingly 1242': the cor- 
respondingarc, o' M, is found by the process prescribed in verse 33 to be 
21° 12'. The figure shows the equation to be subtractive. 

The annexed table presents the results of the calculation of the equa- 
tion of the conjunction (ptghrakarman) for the five planets. 

Results of the First Process for finding the True Places of the Planets. 



Planet. 



Mercury, 

Venus, 

Mars, 

Jupiter, 

Saturn, 



Mean 
Longitude. 



8 18 l3 l3 
8 18 i3 i3 
5 24 3o 57 

2 26 2 14 

3 20 12 3 



Longitude of 
Conjunction. 



Mean 
Commutation. 



4 16 57 22 
10 21 49 47 

8 18 i3 i3 
8 r8 i3 i3 
8 18 i3 i3 



Base- ' Coir. 



7 28 44 9 
2 3 36 34 
2 23 42 16 
5 22 10 59 
4 28 1 10 



Epicycle. ^ 



2938 
3o8o 
34i6 
468 
1820 



Result 
from 



l32 9 1078 
260 l3. 2226 

232 I J 2202 
70 l6 91 

39 3a' 200 



Result 

from 

P. -sine. 



655 
1104 

325 

665 
3ao 



Va 
abl 
Hyp. 



2084 

5o58 
4274 
2774 
3i24 



£quat*n 
of Conj. 



-21 12 

+26 7 

+3i 1 
+ 1 53 
+ 3 4o 



This is, however, only a first step in the whole operation for finding 
the true longitudes of these five planets, as is laid down in the next 
passage. 

43. The process of correction for the apsis (mdnda Jcarman) is 
the only one required for the sun and moon : for Mars and the 
other planets are prescribed that for the conjunction (odighrya), 
that for the apsis {mdnda), again that for the apsis, and that for 
the conjunction — four, in succession. 



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ii. 45.] S&rya-Siddhdnla. 215 

44. To the mean place of the planet apply half the equation 
of the conjunction (^ighraphala\ likewise half the equation of the 
apsis ; to the mean place of the planet apply the whole equation 
of the apsis (Tmndaphala), and also that of the conjunction. 

45. In the case of all the planets, and both in the process of 
correction for the conjunction and in that for the apsis, the equa- 
tion is additive (dhand) when the distance {kendra) is in the half- 
orbit beginning with Aries ; subtractive (rna), when in the half- 
orbit beginning with Libra. 

The rule contained in the last verse is a general one, applying to all 
the processes of calculation of the equations of place, and has already 
been anticipated by us above. Its meaning is, that when the anomaly, 
(mandaJcendra)) or commutation (fighrakendra), reckoned always forward 
from the planet to the apsis or conjunction, is less than six signs, the 
equation of place is additive ; when the former is more than six signs, 
the equation is subtractive. The reason is made clear by the figures given 
above, and by the explanations under verses 1-5 of this chapter. 

It should have been mentioned above, under verse 29, where the word 
hendra was first introduced, that, as employed in this sense by the Hin- 
dus, it properly signifies the position (see note to i. 53) of the u centre " 
of the epicycle — which coincides with the mean place of the planet itself 
— relative to the apsis or conjunction respectively. In the text of the 
Surya-Siddhanta it is used only with this signification : the commentary 
employs it also to designate the centre of any circle. 

Since the sun and moon have but a single inequality, according to the 
Hindu system, the calculation of their true places is simple and easy. 
With the other planets the case is different, on account of the existence 
of two causes of disturbance in their orbits, and the consequent necessity 
both of applying two equations, and also of allowing for the effect of each 
cause in determining the equation due to the other. For, to the appre- 
hension of the Hindu astronomer, it would not be proper to calculate the 
two equations from the mean place of the planet ; nor, again, to calculate 
either of the two from the mean place, and, having applied it, to take 
the new position thus found as a basis from which to calculate the other ; 
since the planet is virtually drawn away from its mean place by the 
divinity at either apex (ucca) before it is submitted to the action of the 
other. The method adopted in this Siddhanta of balancing the two 
influences, and arriving at their joint effect upon the planet, is stated in 
verses 43 and 44. The phraseology of the text is not entirely explicit, 
and would bear, if taken alone, a different interpretation from that which 
the commentary puts upon it, and which the rules to be given later show 
to be its true meaning ; thi3 is as follows : first, calculate from the mean 
place of the planet the equation of the conjunction, and apply the half 
of it to the mean place ; from the position thus obtained calculate the 
equation of the apsis, and apply half of it to the longitude as already 
once equated ; from this result find once more the equation of the apsis, 
and apply it to the original mean place of the planet; and finally, calcu- 
late from, and apply to, this last place the whole equation of the con- 
junction. 



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216 



E. Bwgwy .etc., 



[ii. 45. 



We bare calculated by this method the trne places of the Are planets, 
and present the results of the processes in the following tables. Those 
of the first process have been already given under the preceding pas- 
sage: the application of half the, equations there found to the mean 
longitude gives us the lqngitude once equated as a basis for the next, 
process. 

Results of the Second Process for finding the True Places of the Planets. 



Planet 


Equated 
Longitude. 


Lonrfatde 
of Apaia. 


Equate* 
Anomaly. 


Baee- 

line. 


Corrected 
Epicycle. 


Equation* 
ofApeia. 


Mercury, 

Venus, 

Mars, 

Jupiter, 

Saturn, 


8 7 3 7 

9 l 17 
6 io i 

3 36 59 
3 33 I 


7 IO 38 20 

3 19 53 17 

4 10 3 4o 

5 31 33 19 

7 ad aV34 


II 2 5l 

5 18 35 
10 2 

2 24 a3 
-4 **7 


1 568 
681 
2977 
3430 
2839 


39 5 

11 48 

7a 34 

33 

48 it 


- a . 7 

+ O 33 
-IO 3 
-1-5 5 

•+ r> ao 



Again, the application of half these equations to the longitudes as 
once equated furnishes the data for the third process. The longitudes of 
the apsides, being the same as in the second operation, are not repeated 
, in this table. 

Results <tf the Ttori Process for finding the True Places of the Planets. 



"»*• . 


Equated 
longitude. 


Equated 
Anomaly. 


Baee- J Corrected 
•ino. j fiptcyele. 


Equation 
of Apcli. 




a • f 

8 6 34 

9 I 38 

6. i 
a ^9 3o. 
3 a5 7 u 


11 3 54 

5 18 34 

io* . 5 .3 

• a n-52 
4 1 a7 


• 1 • f 

i5i3 1 29 7 

691 j 11 48 

3fli4 i 7a 33 

34*3 -I 3a 1. 
3932 ] 48 . 9 


• 
-3 3 

+0 33 
-9 3o 
+5 4 
+6 33 



The original mean longitudes are now corrected by the results of the 
third process, to obtain a position!, from which shall be once more calcu- 
lated the equation of the conjunction ; and the application of this to the 
position which furnished it yields, as a final result, the true place of each 
planet. 



Results 


of the Fourth Process for finding the True Places of the Planets. 


Planet. 


Equated 
Longitude. 


BqeMed 
Commuta- 
tion. 


Bane- 
aloe. 


Conr. 
Epicycle. 


Reaolt 

from 

B.-*fne. 


Beanlt 

from 

P.pine. 


Variable 
Hypot*. 


Equation 
efCouj. 


True 
Loofitada, 


Mercury, 

Veuua, 
Mors, 
Jupiter, 
Saturn, 


• • 
8 16 11 
8 18 36 
5 i5 1 
3 1 6 
3 36 45 


e • • 
8 046 
3 3 14 

3 3 12 

5 17 7 

4 31 s8 


3ooo 

3o69 

3433 

766 

3l4l 


132 8 
360 i3 

333 O 

70 37 
39 37 


1 101 
83 18 

3313 

i5o 

336 


616 

1118 

134 

656 
396 


3oa9 
5o67 
3o84 
3786 
3i5r 


-ai ao 
+a5 59 
+33 44 
+ 3 5 
+ 4 17 


7 a4 5i 
9 i4 35 
6 18 45 

3 4 11 

4 1 a 



We cannot furnish a comparison of the Hiiidu determinations of the 
true places of the planets with their actual positions as ascertained by 
our modern methods, until after the subject of the latitude has been dealt 
with : see below, under verses 56-68. 



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li. 45.] 



iif 



The Hindu method of finding th<i Hue longitude* ©f the fit* planets 
whose apparent position is affected by the patella* 6f the earth's ro6tidn 
having thus been fully explained, We #iH profcefcd td indicate, as suc- 
cittctly as possible, the Way in Which the same ptobtem is* solved by the 
great Greek astronomer. The annexed figure (Fig. 6) Will illustrate his 
method : it is taken from those presented in the Syntaxii, but With such 
modifications of form as to make it correspond With the figures £revi- 
<*usly given here : the conditions which it represents a*e only hypothet- 
ical, not according With the actual dements ef any of the planetary 
orbits. 

Let £ be the earth's plaee, and let the circle ApC, described abdut 

£ as a eentre, represent the mean orbit of any plane% £ A being the* 

direettoik of its line of apsides, and £0 that of its conjunction (ptyAM), 

p ^ ealldd by Ptolemy the 1 

apdgtee of its epicycle; 
Let EX be the double 
eccentricity, or the* 
equivalent to the ta* 
dint of the Hindu 
eptycte 6f the apsis' J 
and let EX be hi* 
aected in Q. Then, 
as regards the influ* 
erice" of the eccen' 
trioUjr of the otbit 
upoil the place of tW 
planet, the centre ef 
equable aufula* ft* 
tidn is at X WW the centf* of equal distant is *t Q : th* planet tftt* 
ally describe* the tittl* A' P, of WBidh Q is the defitrt, but at the fcam* 
#*te a* if it We¥e ffldving e^dabty updti the dotted e*ircle, 6f Wfctek f&4 
centre is all X. The angle bi mean ariolnaty, accordingly, Which fin 
efenees pttpeltionalty to the* time, is iXA", but P is the £lanetfi pla^e, 
P E A the title Utatoflaty, and £ P X the equation of friad. The valut 
of EPX fo dbtaified by a process ftnalagans to that described abofe\ 
under tflfee 69 (pp. 210, 211); EB and BX, and QD and DX, JW» 
first found; then DP, Which, by suUractirig DX, gives XP; IP 
added t6 BX gives BP> and frdiii B t and BEl is derived fcPB, the 
equation required; subtract this froth PXA, and th« rtfrosande? H 
BE A, the planet's true distance^ from the apsis. About P descried 
the epicycle of the conjunction, and draW the radius PT parallel 
td EC: then T is tha planet's place in tb6 apicjrele, p its" apparent 
position hf the mean orbit, and TE P the eauMion of the epicycle; 6t 
ef the conjunction. In order to arrive at the value of this equation, 
Ptolemy first finds that of SEE, the corresponding angle When the 
Centre of the epicycle is placed at R, at the mean distance B B, 6# 
radius, from IS : he then diminishes it by a complicated process, into 
the details of which it is not necessary here to enter, and which, as 
be himself acknowledges, is not strictly accurate, but yields results suffi- 
ciently near to the truth. The application of the equation thus obtained 




VOL. VI, 



28 



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218 E. Burgest, eft, [ii. 45. 

to the place of the planet at already onoe equated gives the final result 
•ought for, itt geocentric place. 

In the case of Mercury, Ptolemy introduces the additional supposition 
that the centre of equal distances, instead of being fixed at Q, revolves 
in a retrograde direction upon the circumference of a circle of which 2 
is the centre, and X Q the radius. 

After a thorough discussion of the observations upon which his data 
and his methods are founded, and a full exposition of the latter, Ptolemy 
proceeds himself to construct tables, which are included in the body of 
his work, from which the true places of the planets at any given time 
may be found by a brief and simple process. The Hindus are also ac- 
customed to employ such tables, although their construction and use are 
nowhere alluded to in this treatise. Hindu tables, in part professing to 
be calculated according to the Surya-Siddh&nta, have been published 
by Bailly (Trait* de l'Astr. Ind. et Or., p. 335, etc.), by Bentlev (Hind. 
Ask, p. 219, etc.), by Warren (K&Ia Sankalita, Tables), by Mr. HoiaiDg- 
ton (Oriental Astronomer, p. 61, etc.), and, for the sun and moon, by 
Davis (As. Res., ii. 255, 256). 

We are now in a condition to compare the planetary system of the 
Hindus with that of the Greeks, and to take note of the principal re- 
semblances and differences between them. And it is evident, in the first 
place, that in all their grand features the two are essentially the same. 
Both alike analyze, with remarkable success, the irregularities of the 
apparent motions of the planets into the two main elements of which 
they are made up, and both adopt the same method of representing and 
calculating those irregularities. Both -alike substitute eccentric circles 
for the true elliptic orbits of the planets. Both agree in assigning to 
Mercury and Venus the same mean orbit and motion as to the sun, and 
in giving them epicycles which in fact correspond to their heliocentric 
orbits, making the centre of those epicycles, however, not the true, but 
the mean place of the sun, and also applying to the latter the correction 
due to the eccentricity of the orbit Both transfer the centre of the 
orbits of the superior planets from the sun to the earth, and then assign 
to each, as an epicycle, the earth's orbit; not, however, in the form of 
an ellipse, nor even of an eccentric, but in that of a true circle ; and 
here, too, both make the place of the centre of the epicycle to depend 
upon the mean, instead of the true, place of the sun. The key to the 
whole system of the Greeks, and the determining cause both of its nu- 
merous accordances with the actual conditions of things in nature, and 
of its inaccuracies, is the principle, distinctly laid down and strictly ad- 
hered to by them, that the planetary movements are to be represented 
by a combination of equable circular motions alone, none otner being 
deemed suited to the dignity and perfection of the heavenly bodies. By 
the Hindus, this principle is nowhere expressly recognized, so far as we 
are aware, as one of binding influence, and although their whole system, 
no less than that of the Greeks, seems in other respects inspired by it, 
it is in one point, as wc shall note more particularly hereafter, distinctly 
abandoned and violated by them (see below, under vv. 50, 51). We 
cannot but regard with the highest admiration the acuteness and in- 
dustry, the power of observation* analysis, and deduction of the Greeks, 



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ii: 45.] Sarya-Siddh&nta. 219 

that* hampered by false assumptions, and' imperfectly provided wiA 
instruments, they were able to construct a 'science containing so mucfc 
of truth, and serving as a- secure basis for the improvements of after 
time: whether we pay the same tribute to the genius of the Hindu will 
depend upon whether we consider him also, like all the rest of the world, 
to have been the pupil of the Greek in astronomical science, or whether 
we shall believe him to have arrived independently at a system so 
closely the counterpart of that of the West. 

The differences between the two systems are much less fundamental 
and important. The assumption of a centre of equal distance different 
from that of equal angular motion — and, in the case of Mercury, itself 
also movable — is unknown to the Hindus : this, however, appears to be 
an innovation introduced into the Greek system by Ptolemy, and un- 
known before his time ; it was adopted by him, in spite of its seeming 
arbitrariness, because it gave him results according more nearly with his 
observations. The moon's evection, the discovery of Ptolemy, is equally 
Wanting in the Hindu astronomy. As regards the combined application 
of the equations of the apsis and the conjunction, the two systems are 
likewise at variance. Ptolemy follows the truer, as well as the simpler, 
method : he applies first the whole correction for the eccentricity of the 
orbit, obtaining as a result, in the case of the superior planets, the 
planet's true heliocentric place ; and this he then corrects for the paral- 
lax of the earth's position. Here, too, ignorant as he was of the actual 
relation between the two equations, we may suppose him to have been 
guided by the better coincidence with observation of the results of his 
processes when thus conducted. The Hindus, on the other hand, not 
knowing to which of the two supernatural beings at the apsis and con* 
junction should be attributed the priority of influence, conceived them 
to act simultaneously, and' adopted the method stated above, in verse 44; 
of obtaining an average place whence their joint effect should be calcic- 
kited. This is the only point where they forsook the geometrical method^ 
and suffered their theory respecting the character of the forces produ- 
cing the inequalities of motion to modify their processes and results. 
The change of dimensions of the epicycles is also a striking peculiarity 
of the Hindu system, and to us, thus far, its most enigmatical feature. 
The virtual effect of the alteration upon the epicycles themselves is to 
give them a form approximating to the elliptical. But, although the 
epicycles of the conjunction of the inferior planets represent the proper 
orbits of those planets, and those of the superior the orbit of the earth, 
it is not possible to see in thia alteration an unconscious recognition -of 
the principle of ellipticity, because the major axis of the quasi-ellipae— 
or, in the case of Jupiter and Saturn, the minor axis — is constantly 
pointed toward the earth. Its effect upon the orbit described by the 
planet is, as concerns the epicycle of the apsis, to give to the eccentric 
circle an ovoid shape, fattened in the first and fourth quadrants, bulging 
in the second and third : this is, so far as it goes, an approximation to- 
ward Ptolemy's virtual orbit, a circle described about a centre distant 
from the earth's place by only half the equivalent of the radius of the 
Hindu epicycle (the circle A' P in figure 6) : but the approximation 
seems too distant to furnish any hint of an explanation. A diminution 



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# &wgw< fo x 



pi. 46- 



tf Ifa tpfttird* ffo'tlMt * CDjrearonifing oMxnjsnrtta* ef the *qnatfon, 
o*rfying the pUwt forwa*4 where tine equation i* aubfcreethre, and back* 
wa>rd wnere it ift edolitive : l>Ht we hardly feel justified in, assuminff that 
jt |p to b* regarded a* *n. empirical correction, applied to make thei*- 
stilts of osculation agree mere needy with those of observation, because 
jtft W&onnt *a4 pla>ce stand in nq relatf** which we h**e been *Me t* 
tcac$ to the true eieinente of the. planetary orbits, nor is the accnra/gn 
of either the Hindu calculation* pr observations so great a* to/mate 
such slight corrections of appreciable importance. We am cmnpeUvi 
to leave, the solution of this difficulty, if it shall prove soluble, to later 
investigation^ and a, move extended comparison of the different tejttp 
feeojcs of Hindu astronomical science. 

M. regards the numerical value of the elements adopted by the two 
smtyms-^eir xoutnsj relation, and their respective relations to the true 
tlern.enis established by modern science, are exhibited in the annexed 
tftblq, fhe, first part of it presents the comparative dimensions of tie 

eftoetary orbits, or the value of the radius, of each in terms of that of 
e earth's orbi£ In the case of Mercury and Venus, this ia represent*} 
by the relation of the radius of the epicycle (of the conjunction) to 3hat 
of the orbit; in the case of the superior planets, by that of- the raxtnsj 
of the. orbit te. the radius of the epicycle. For the Hindu system it 
Wsa necessary to give two values in every case, derived respectively frou) 
the greatest and least dimensions of the epicycles. Such a relative oV 
termination of the moon's orbit* of course, could noi^ be obtained; ft* 
absolute dunenaione will be found staged late? (see under iv«>8 and x& 
§4). The second part of the table gives, a* the fairest practicable eota* 
parson of the value* assigned! by each aystem to the ecoentricities, the 
greatest equations, of the centre, for Mercury and Venus* however, the 
ancient e,nd modern determination of these equations are not at aJJ 
S^P*r*W<H tJh latter giving* their actual heliocentric amount, the fes* 
Tfiev their apparent vaJLuo, as eee* fron* the earth. 

JMoJjt* 3imen$fo*t emd X&mtrieitiee cf the Planetary Orbit*, awarding 
to Different Authorities. 



rjiii.*.'.. . , j. 


ita<aaa of tba Orbit 


3«seta« Eqaatioo of the Ctutr*. 


tt*a*t. 

\ ■ 


8urya-Si 
evrnqaad. 


ddbanta. 
oddqntd 


Ptota»y« 


Moean* 


Suits- 
Sidddh&nta. 


Fteltnfv 


Hodta*. 


Sim, 


1.0000 


1*0000 


1.0000 


t*oooo 


a 10 3i 


rl 23 


1 55^ 


Moon, 


• • . « • 


. .. • . 




. • .-. . 


5 a 46 


S I 


17* ^3. 


Mtwury*. 


.36o4 


•3667 


.3 7 5o 


.38 7 i 


4 27 35 


2 52 


a3 4o.43 


Venus, 


.737® 


.732 a 


•7io4 


■7233 


1 45 3 


3 23 


47 M 


Mars. 


i.5i39 


i.55i3 


1.5190 


r 1.5337 


11 3a. 3 


11 3a 


to 4t 33 


Juaiter ( 


5. Mae 


5>ODOO 


5.a ij4 


5.aoa8 


5 5 58 


16 


'S3i vi*. 


Batura, 


9.s3oo 


9.0000 


9i3o8 


9-S389 


7 39 3a 


6- 3a 


;6 a£ ra; 



46. Multiply the daily motion (bhuktt) q{ a planet by the atm'a 
result from the ^aae-sioe (hdhuphctfa)) and divide bjr the ^UBtbqr 
pf minute* in a circle (bkqeajcra) j the resmJt^ in ¥Wutes> epp]j 
tQthepUHei^t^ue pla<w, » tta sejrgi ^eci^ja^ ^*<&w$Af 

— 1 appUeai tq ttaanj^ \ .; ;0^;v ■---.- c 



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it 49,] S&ryfr&44h$nta. 22X 

By ibis rale, allowance is iniide fur that part of the equation <rf tbne, 
or pf the difference betproen mean and apparent soiar tin**, which is da*' 
to the difference between the sun's mean and true places. The instru- 
ments emnloyed bjr the Hindus in measuring time are described, very 
briefly sot insufficiently, in the thirteenth chapter of this work : in aft 
probability the gnonjeu and shadow was, that meat relied upon ; at any 
rate, they can have had no means of keeping mean time with any acee> 
racy, ana H appears franr this passage that apparent time alone is r*> 
garded as ascertainable direody. Now if the sun moved in the eqnv 
nectial instead of in the ecliptic, the interval between the passage, of his 
giean and his true place across the meridian would be the same part of 
a day, as the difference, of the two places is of a circle : bcnoe the pr*> 
portion upon which the nde in the text is founded '< as the anfsbe* of 
<ninut«a in a circle is to that in the son's equation (which is the same 
with his "result from the base-sine:" see above, v. 89), so-ia the wbcA* 
©>fly motion of any planet to its motion during the interval. Atti 
since, when the son is in advance of his true place, he cojtej l**et t* 
tho meridian, the planet moving on during the interval, and the re*ect% 
the result is additive to the planet's place, or subtract! ve from it, aooord- 
Pga* the ann's equation is additive or snbtractive, 

The Other source of difference between true and apparent (foe, the 
difference in the daily increment of the arcs of the etfiptie, in wbitb 
the sun moves, and of those of the equinoctial, which are the measure* 
of time, is not taken account of in this treatise. This. is {h* more 
strange, as that difference is, for soma other puipeaes, calculated and 
ftttawed for. 

At the tea lor which we have ascertained above the true places of 
the planets, the sun i» ao pear the. perigee, and his equate* of place is 
sot small, thafc it renders necessary ne> modification of the places aa 

S'ven; even the moan more* but a small fraction of a second during 
$ interval between mean and apparent midnight 
By oAtf Jt*v W w4 fa t&* verse, we axe to understand, of epurao, not 
th.e mean, but the actual, daily motion of the planet : the commentary 
aJsp gives the word this interpretation. How the actual rate of motioj* 
U fc*n4 *t any given time, is taught in the next passage, 

4T. From the mean dairy motion of the moon, subtract the* 
daily motion of its apsis (manda), and, having^ireated the differ- 
ence in tho manner prescribed by tho next rufe) apply the, result*, 
as an additive, or suotractive equation, to the daily motion. 

4$. The equation of a planet s daily motion. is to be calculate^ 
life$ the> place of the planet in the propess for the apsis ; multi- 
ply the daily motion by tb* differenca of tabular sines corre- 
sponding tq th* baseline (tfcwjjtf) of aa<wndy, wod then divid* 
by two hundred a,nd. twenty-five ; 

4Q. Multiply the result by the corresponding epxryole of the> 
aipsn (mndapqrklhi), and divide by the number of degrees ii) a 
Circle (bhagaruz) ; tho result, in mitrotea, is additive when in the 
fal£orbit beginning with Qaneer, aad eubtractivo vrfceh in that 
beginning with Onprieorn. •'- : L - : - " : * ■• • : 



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222 EJBwrgm, etc., [ii. 49- 

Only the effect of the apsis upon the daily rate of motion is treated 
of in these verses; the farther modification of it by the conjunction is 
the subject of those which succeed. 

Verse 47 is a separate specification under the general rule given in 
tlte following verse, applying to the moon alone. The rate of a planet's 
motion in its epicycle being equal to its mean motion from the apsis, or 
its anomalistic potion, it is necessary in the case of the moon, whose 
apsis has a perceptible forward movement, to subtract the daily amount 
of tliis movement from that of the planet in order to obtain the daily 
rate of removal from the apsis. 

In the first half of verse 48 the commentary sees only an intimation 
that, as regards the apsis, the equation of motion is found in the saipe 
general method as the equations of place, a certain factor being multi- 
plied by the circumference of the epicycle and divided by that of the 
ortnt. Such a direction, however, would be altogether trifling and super- 
fluous, and not at all ia accordance with the usual compressed style of 
the treatise; and moreover, were it to be so understood, we should lack 
any direction as to which of the several places found for a planet in the 
process for ascertaining its true place should be assumed as that for 
phich this first equation of motion is to be calculated. The true mean- 
ing of the line, beyond all reasonable question, is^ that the equation is 
to be derived from the same data from which the equation of place for 
the apsis was finally obtained, to be applied to the planet's mean posi- 
tion, as this is applied to its mean motion; from the data, namely, of 
the third process, as given above. 

The principle upon which the rule is founded. may be. explained as 
follows. The equation of motion for any given time is evidently equal 
to the amount of acceleration or of retardation effected during that time 
by the influence of the apsis. Thus, in Fig. 3 (p. 208), m*, the sine of 
a' m, is the equation of motion for the whole time daring which the 
centre of the epicycle has been traversing die arc AM. If that arc. 
and the arc a'm, be supposed to be divided into any number of equal 
portions, each equal to a day's motion, the equation of motion for each 
successive day will be equal to the successive increments of the sines of 
the increasing arcs in the epicycle ; and these will be equal to the suc- 
cessive increments from day to day of the sines of mean anomaly, re- 
duced to the dimensions of the epicycle. But the rate at which the 
sine is increasing or decreasing at any point in the quadrant is approxi- 
mately measured by the difference of the tabular sines at that point : 
and as the arcs of mean daily motion are generally quite small— being, 
except in the case of the moon, much less than 3° 45', the unit of the 
table— we may form this proportion : if, at the point in the orbit occu- 

Sied by the planet, a difference of 8° 45' in arc produces an increase or 
ecreaae of a given amount in sine, what increase or decrease of sine 
will be produced by a difference of arc equal to the planet's daily 
motion ? or, 225 : diff. of tab. sines : : planet's daily motion : correspond- 
ing dfff. of sine. The reduction of the result of this proportion to the 
dimensions of the epicycle gives the equation sought. 

We will calculate by this method the true daily motion of the moon 
at the time for which her true longitude has been found above. 



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ii. 51.] SOrya-SiddMnta. 223 

Moon's mesa daily motion (i SOX . . 79©' 35" 

deduct daily motion of apsis (L M), 6 4i 

Moon's mean anomalistic motion, 783 54 • 

From the process of calculation of the moon's true place, given above, 
we take 

Moon's mean anomaly, io* 18* 46^15" 

Bine of anomaly (bhvyitfyd), aa66' 

From the table of sines (ii. 15-27), we find 

Corresponding diffsience of tabular sines, • it4' 

Hence the proportion 

225' : *74' : : 7*3' 54" : 606' i3" 

shows the increase of the sine of anomaly in a day at this .point to be 
606' 13". The dimensions of the epicyole were found to be 81° 47f. 
Hence the proportion • 

36c*:3i°47'::6o6' i3":53'3i" 

give us the desired equation of motion, as 58' 81". By Terse 49 it is 
subtractive, the planet being less than a quadrant from the apsis, or its 
anomaly being more than nine and : tess than three signs. Therefore, 
from the 

Moon's mean daily motion, 79c/ 35" 

subtract the equation, 53 3i 

Moon's true daily motion at given time, 737 4 

The roughness of the process is well illustrated by this example* 
Had the sine of anomaly been but V greater, the difference of tines 
would have been IO* less, and the equation only about 50'. 

The equation of the sun's motion, calculated in a similar manner, is 
found to oe +2' 18", and his true motion 61' 26". 

The corrected rate of motion of the other planets will be given under 
the next following passage. 

50. Subtract the daily motion of a planet, thus corrected for 
the apsis (manda\ from the daily motion of its conjunction 
(dghra) - then multiply the remainder by the difference between 
the last hypothenuse and radius, 

. 51. And divide by the variable hypothenuse {cola Jcai*na): the 
result is additive to the daily motion when the hypothenuse is 
greater than radius, and subtractive when this is lees; if, when 
subtractive, the equation is greater than the daily motion, deduct 
the latter from it, and the remainder is the daily motion in a 
retrograde (vakra) direction. 

The commentary gives no demonstration of the rule by which we are 
here taught to calculate the variation of the rate of motion of a planet 
occasioned by the action of its conjunction : the following figure, how- 
ever (Fig. 7), will illustrate the principle upon which it is founded. 



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tb* E; Buy**, «afc, [& *i, 

As in * previous figure (Fig. 5, p. 218), CMM y represents tht Wean 
orbit of ft planet, £ the earth, and M the ttlaneft mean JjonitSdtt, at a 
given time, relative to its conjunction, C : the cirele described abort M 
is the epicycle of the conjunction : it is drawn, in the figure, of the 
relative dimensions of that assumed 
for Mars. Suppose M' M to be the ^^^^**si 
amount of motion of the centre of r~ 
the epicycle, or the (equated) mean 
synoaicai motion of the planet, 
during one day ; m 1 m is the art Of 
the epicycle traversed by the planet 
in the same time. As the amount 
of daily synodical motion is in every 
case small, these arcs are necessarily 
greatly exaggerated la the ngita, 
being made about twenty-four times 
too great for Mars. Had the planet 
remained stationary in the epicycle 
at n%' while the centre of the epi- 
eyele moved from M' to M, its plaeo 
at the giten ttee would be at -#; - 
having moved to m, it is seen at I : 
fccnoe $ t is the equation of daily motion, of. which, it is required to 
ascertain thetaioe, Pi^o^e Em'^v*"^ 
join «rt % ; from M draw M o at right angles to Em. Theft, SiMe Ihe are 
mm' is very small, the angles Bm» and Enm/ as also Mmm' and 
M m'm, may be regarded as right angles ; Mm b and* n m m! art there- 
to* *q«al* each being the cetnfleaett «f Bee*', awd 4b* triangles 
nmW an4Mm#artilUnllar. HeACf 

Mm*mdlfOtfjt^:mlt 

But EM:Mmi:MM'>mm' 

Hdnce, by combining tenia, l»M:m*:iMM'jm* 

But **:E<::m»3£j» 

therefore, siaee EM eeuaWBO j A ,-h A ».wu'.1J** 
by again eomUong/ f "<**!. MM .S«e 

and, reducing the ftfoportkra to « equation, rt> the feftiftd tquattai 
of motion, equals M M', the equated mean synodical motion in a day, 
multiplied by m o, and divided by B ^ the variable hypothenuse. This, 
however, is not precisely the rule given above; for >n the text of this 
iKeMh&fita, m ^ the eWereriee between the variable bypothenute and 
radius, is substituted for mo, as if the two were virtually equivalent: a 
highly inaccurate assumption, since they differ from one another by the 
versed sine, o *, bt the equation of the conjunction, M t t which equation 
is^sometimes as much as 40° : and indeed, the commentary, contrary to 
its usual habit of obsequiousness to the inspired text with which it has 
|to deal, rejects this assumption, and says, without even an apology for* 
the liberty it is taking, that by the word a radius" in verse 50 is to be 
understood the ooaine {kotfy&) of the second equation of the conjunction* 




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ft 51,] 



8&rya-S£ddhdrita. 



225 



la itinstration of fne' rule, we will . calculate the true rate of daily 

-motion of. the. planet Mare, at the same time for which the previous 

(calculations hare been made. 

i • By the process already illustrated under the preceding passage, the 
equation of Mars's daily motion for the effect of the apsis, as derived from 
the data of the third process for ascertaining hi& true place, is found to 

l ^y£§' 4*1", the difference of tabular .sines being! SI'. Accordingly,. 

from the mean daily motion of Mira (t. M), 3i' 26" 

' deduct fbe equation for the apeb, .' 3 4i 

Mare's equated daily motion, 

^Kow, to find the equated daily synodical motion', 

«*• . .. from the daily motion of Man's; coygunction (tha mm), 
|^ , deduct his equated daily motion, 

%i' V*tyf£ equated daily synodical motion, 



27 45 



V 8" 
27 45 

3i a3 



g The Variable bypothenuse used in the last process fox finding the true 

ftgHace was $964'; its exeees abort radius is 546'. The proportion 

r 3964':546 / ;:3* / .a3' , :4'x6" 

shows, then, that the eauatkm of motion due- to the conjunction at the 
given time is 4' 18". Since the hypothennse.is greater than radius— 
fiiairk txrsay, sinoe the planet it in ta* haiforbit ia which the influence 

i«£ the ccarjuaction k sa^eratire^t-the ^equation is additive. Therefore, 

\"* '" to MaiVs equated daily motion, vf &" 

' a3d the equation for the conjunction, 4 18 

- j lta^»4rue daily m«tbn at the f^ettt^nei 3a ,3. 

In this calculation we have followed the rule stated in the text : had 
we accepted the atnendmen4r of- the commentary, and, in finding the 
second term : o£. our proportion,* .substituted for radius the cosine of 
33° 44', the resulting aquation would have been more than doubled, 
becoming 8' 51", instead of 4' 18"; this happening to be a case where 
the difference is nearly as great as possible. We have deemed it best, 
however, in making out the corresponding results for all the toe planets, 
as presented in the annexed table, to adhere to the directions of the 
text iteelfc The inaccuracy, it may be observed, is greatest when the 
equation of motion is least, and the contrary; so that, although some- 
times very large relatively to the equation, it never comes to be of any 
great importance absolutely, 

Jfautli of (he Processes for finding the True Baity Motion of the Planed. 



1, . . tf r . 

1 • Fltaei. 


Diff 
ofiinet. 


oTA|»it. 


Equated 
Motion. 


Equated [ Equation 
Syaod.MoUoa. of Conjunction. 


True 
Motiea, 


* ^Mercury, 


2o5 


-4 21 


54 47 


190 45 


-25 45 


» r* 
+29 2 


* ' Yenua, 


219 


+1 53 


61 1 


35 7 


+ 1! 17 


+72 1*- 


Mara, 


l3l 


-3 4i 


27 45 


3i 23 


+ 4 18 


+32 3 


Jupiter, 


3 7 


-0 4 


4 55 


54 i3 


-12 4t 


-7 46 


Saturn,- 


119 


+0 8 


2 8 


. 57 a 


-» 5. M 


-3 .3. 



VOL. VI. 



29 



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226 E. Burgess, etc., [n. ai- 

The final abandonment by the Hindus of the principle of equable cir- 
cular motion, which lies at the foundation of the whole system of eccen- 
trics and epicycles, is, as already pointed out above (under vv. 43-45), 
distinctly exhibited in this process: m'm (Fig. 7), the arc in the epi- 
cycle traversed by the planet during a given interval of time, is no fixed 
and equal quantity, but is dependent upon the arc M' M, the value of 
which, having suffered correction by the result of a triply complicated 
process, is altogether irregular and variable. This necessarily follows 
from the assumption of simultaneous and mutual action on the part of 
the beings at the apsis and conjunction, and the consequent impossibility 
of constructing a siugle connected geometrical figure which shall repre- 
sent the joint effect of the two disturbing influences. By the Ptolemaic 
method the principle is consistently preserved : the fixed axis of the 
epicycle (see Fig. 6, p. 217), to the revolution of which that of the 
epicyclo itself is bound, is arPX; and as the angle *PT, like *XA", 
increases equably, the planet traverses the circumference of the epicycle 
with an unvarying motion relative to the fixed point x ; although the 
equation is derived, not from the arc *T, but from <T, the equivalent of 
CR, its pert ex varying with the varying angle EPX. 

In case the reverse motion of the planet upon the half-circumference 
of the epicycle, within the mean orbit is, when projected upon the orbit* 
greater than the direct motion of the centre of the epicycle, the platfef 
will appear to more backward in its orbit, at a rate equaj to the mvm 
of the former over ^he latter motion. This is, as the last table ahowa, 
the case with Jupiter and Saturn at the given time. The subject of thus 
retrogradation of the planets is continued and completed in tfee next 
Mowing passage. 

52. When at a great distance from its conjunction ($ghrocca\ 
a planet, having its substance drawn to the left and right by 
slack cords, comes then to have a retrograde motion. 

53. Mars and the rest, when their degrees of commutation 
(kendra), in the fourth process, are, respectively, one hundred 
and sixty-four, one hundred and forty-four, one hundred and 
thirty, one hundred and sixty-three, one hundred and fifteen, 

54. Become retrograde (vakrin): and when their respective 
commutations are equal to the number of degrees remaining 
after subtracting those numbers, in each several case, from a, 
whole circle, they cease retrogradation. 

55. In accordance with the greatness of their epicycles of the 
conjunction {gighraparidJii), Venus and Mars cease retrograding 
in the seventh sign, Jupiter and Mercury in the eighth, Saturn 
}n the ninth. 

The subject of the stations and retrogradations of the planets is 
rather briefly and summarily disposed of in this passage, although 
treated with as much fullness, perhaps, as is consistent with the general 
method of the Siddhanta. Ptolemy devotes to it the greater part of 
the twelfth book of the Syntaxis. 



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ii. 55.] S&rya-Siddhdnta. $&1 

The first verse gives the theory of the physical cause of the phenome- 
non : it is to be compared with the opening verses of the chapter, 
particularly verse 2. We note here, again, the entire disavowal of the 
system of epicycles as a representation of the actual movements of the 
planets. How the slackness of the cords by which each planet is 
attached to, and attracted by, the supernatural being at its conjunction, 
furnishes an explanation of its retrogradation which should commend 
ftself as satisfactory to the mind even of one who believed in the super- 
natural being and the cords, we find it very hard to see, in spite of the 
explanation of the commentary : it might have been better to omit 
verse 52 altogether, and to suffer the phenomenon to rest upon the 
simple and intelligible explanation given at the end of the preceding 
verse, which is a true statement of its cause, expressed in terms of the 
Hindu system. The actual reason of the apparent retrogradation is, 
indeed, different in the case of the inferior arid of the superior planets. 
As regards the former, when they are traversing the inferior portion of 
their orbits, or are nearly between the sun and the earth, their helio- 
centric eastward motion becomes, of course, as seen from the earth, 
westward, or retrograde ; by the parallax of the earth's motion in the 
same direction this apparent retrogradation is diminished, both in rate 
and hi continuance, but is not prevented, because the motion of tho 
inferior planets is more rapid than that of the earth. The retrograda- 
tion of the superior planets, on the other hand, is due to the parallax of 
the earths motion in tfoe same direction wl^eu between. them- and the? 
jurijittHf ne lejssen^d by thefr. wfv in oHioir ijh- theft- Orbits,* although" not 
iftnfre awy^ftfrfrltqE^ Aotioniif^ess'-Vaptd thauthaf 

of the earth. But, in the Hindu system, the revolution oftfci pfaielHtf 
the^^pjeycl* qX tjie .conjunction represents, in Ahe_one..ca§e the proper 
inotfpri of >fche planet,- itf "the other; that of the earth, reversed; herice, 
whenever its apparent amount, in a contrary direction, exceeds that of 
the movement of the centre of the epicycle— ^which is, in the one case, 
that of the earthy in the other, that of the planet itself— retrogradation 
Is the necessary consequence. 

- Verses 53-55 contain a statement of the limits within which retro* 
gradation takes place. The data of verse 53 belong to the different 
planets in the order, Mars, Mercury, Jupiter, Venus, and Saturn (see 
above, under i. 51, 52). That is to say, Mercury retrogrades, when his 
equated commutation, as made use of in the fourth process for finding 
his true place (see above, under vv. 43-45), is more than 144° and less 
than 216°; Venus, when her commutation, in like manner, is between 
163 p and 197°; Mars, between 164° and 196°; Jupiter, between 130° 
and 230°; Saturn, between 115° and 245°. These limits ought not, 
however, even according to the theory of this Siddhanta, to be laid 
down with such exactness ; for the precise point at which the subtractive 
equation of motion for the conjunction will exceed the proper motion 
of the planet must depend, in part, upon the varying rate of the latter 
as affected by its eccentricity, and must accordingly differ a little at 
different times. We have not thought it worth while to calculate the 
amount of this variation, nor to draw up a comparison of the Hindu 
with th* Greek and the modern determinations of the limits of retro- 



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238; R Burgess, etc., [ii. 56- 

gradation, ftince-these are dependent for their correctness upon the accu- 
racy, of the elements assumed, and the processes employed, both of 
which have been already sufficiently illustrated. 

. The last verse of the passage adds little to what had been already 
said, being merely a repetition, in other and less precise terms, of the 
specifications of the preceding verse, together with the assertion of a 
relation between the limite of retrogradation and the dimensions of the 
respective epicycles ; a relation which is only empirical, and which, as 
regards Venus and Mars, does not quite hold good. 

' 56. To the nodes of Mars, Saturn, and Jupiter, the equation 
of the conjunction is to be applied, as to the planets themselves 
respectively; to those of Mercury and Venus, the equation of 
the apsis, as found by the third process, in the contrary direction. 

„ 57. The sine of the arc found by subtracting the place of the 
node from that of the planet — or, in the case of Venus and 
Mercury, from that of the conjunction — being multiplied by the 
extreme latitude, and divided by the last hypothenuse — or, in 
the oase of the moon, by radius— gives the latitude (vikshepa). 

58. When latitude and declination (apakrama) are of like 
direction, the declination (krdnti) is increased by the latitude ; 
when of different direction, it is diminished by it, to find the 
ftue (spaskta) declination : that of the sun remains as already 
determined'. 

How to. find the declination of a planet at any given point in the 
ecliptic, or circle, of declination (krdntivrtta), was taught us in verse 28 
above, taken in connection with verses 9 and 10 of tne next chapter: 
fyere we have stated the method of finding the actual declination of any 
planet, as modified by its deviation in latitude from the ecliptic. 

The process by which the amount of a planet's deviation in latitude 
from the ecliptic is here directed to be found is more correct than might 
have been expected, considering how far the Hindus were from compre- 
hending the true relations of the. solar system. The throe quantities 
employed as data in the. process are, first, the angular distance of the 
planet from its node; second, the .apparent value, as latitude, of its 
greatest removal from the ecliptic, when seen from the earth at a mean 
distance, equal to the radius of its mean orbit; and lastly, its actual 
distance from the earth. Of these quantities, the second is stated for 
eacjbf planet in the concluding verses of the first chapter; the third is 
correctly represented by the variable hypothenuse (cala karna) found in 
the fourth process for determining the planet's true place (see above, 
under vv, 43-45) ; the first is still to be obtained, and verse 56 with the : 
first part of verse 57 teach the method of ascertaining it. The princi- 
ple of this method is the same for all the planets, although the state* 
ment of it is so different; it is, in effect, to apply to the mean place of 
the planet, before taking its distance from the node, only the equation 
of the apsis, found as the result of the third process. In the case of 
the superior planets, this method has all the correctness which the: 
Hindu system admits; for by the first three processes of correction is 



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ii. 68.] S&rya-Siddh&nia. 229 

found, as nearly as the Hindus are able to find it, the true heliocentric 
place of the planet, the distance from which to the node determines, of 
course, the amount of removal from the ecliptic. Instead, however, of 
taking this distance directly, rejecting altogether the fourth equation, 
that for the parallax of the earth's place, the Hindus apply the latter 
both to the planet and to the node ; their relative position thus remains 
the same as if the other method bad been adopted. 

Thus, for instance, the position of Jupiter's node upon the first of 
January, 1860, is found from the data already given above (see i. 41-44) 
to be 2* 19° 40 7 ; his true heliocentric longitude, employed as a datum in 
the fourth process (see p. 218), is 3« 1° 6'; Jupiter's heliocentric dis- 
tance from the node is, accordingly, 11° 26'. Or, by the Hindu method, 
the planet's true geocentric place is 3* 4° 11', and the corrected longi- 
tude of its node is 2" 22° 45'; the distance remains, as before, 11° 26'. 

In the case of the inferior planets, as the assumptions of the Hindus 
respecting them were farther removed from the truth of nature, so their 
method of finding the distance from the node is more arbitrary and less 
accurate. In their system the heliocentric position of the planet is rep- 
resented by the place of its conjunction (ftffhra), and they had, as is 
shown above (see ii. 8), recognized the fact that it was the distance of 
the latter from the node which determined the amount of deviation from 
the ecliptic. Now, in ascertaining the heliocentric distance of an infe- 
rior planet from its node, allowance needs to be made, of course, fox the 
effect upon its position of the eccentricity of its orbit But the Hindu 
equation of the apsis is no true representative of this effect : it is calcu- 
lated in order to be applied to the mean place of the sun, the assumed 
centre of the epicycle — that is, of the true orbit ; its value, as found, is 
geocentric, and, as appears by the table on p. 220, is widely different 
from its heliocentric value; and its sign is plus or minus according as 
its influence is to carry the planet, as seen from the earth, eastward or 
westward ; while, in either case, the true heliocentric effect may be at 
one time to bring the planet nearer to, at another time to carry it farther 
from, the node. The Hindus, however, overlooking these incongruities, 
and having, apparently, no distinct views of the subject to guide them 
to a correcter method, follow with regard to Venus and Mercury what 
seems to them the same rule as was employed in the case of the other 
planets — they apply the equation of the apsis, the result of the third 
process, to the mean place of the conjunction ; only here, as before, by 
an indirect process : instead of applying it to the conjunction itself, they 
apply it witn a contrary sign to the node, the effect upon the relative 
position of the two being the same. 

Thus, for instance, the longitude of Mercury's conjunction at the 
given time is (see p. 214) 4» 16* bV ; from this subtract 2° 2', the equa- 
tion of the apsis found by the third process, and its equated longitude 
is 4* 14° 65': now deducting the longitude of the node at the same 
time, which is 20° 41', we ascertain the planet's distance from the node 
to be 8 s 24° 14'. Or, by the Hindu method, add the same equation to 
the mean position of the node, and its equated longitude is 22° 48'; 
subtract this from the mean longitude of tne conjunction, and the dk- 
tance is, as before, 3# 24° 14'. 



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280 



& Burgess^ etc,, 



[iu.58- 



The planet's distance from the node being determined, its latitude 
would be found by a process similar to that prescribed in verse %& of 
this chapter, if the earth were at the centre of motion ; and that rule is 
accordingly applied in the case of the moon ; the proportion being, as 
radius is to the sine of the distance from the node, so is the sine of ex- 
treme latitude (or the latitude itself, the difference between the sine and 
the arc being of little account when the arc is so small) to the latitude 
at the given point. In the case of the other planets, however, this pro- 
portion is modified by combination with another, namely : as the .last 
variable hypothenuse (cala karna), which is the line drawn from the 
earth to the finally determined place of the planet, or its true distance, 
is to radius, its mean distance, so is its apparent latitude at the mean 
distance to its apparent latitude at its true distance. That is, with 

R : sin nod. disk : : extreme lat : actual lat at dist B 
combining var.hyp: R :: lat. at dist R: lat. at true dist 
we have var. hyp : sin nod. dist. : : extreme lat. : actual lat at true dist 
which, turned into an equation, is the rule in the latter half of v. 57. 

The latitude, as thus found, is measured, of course, upon a secondary 
to the ecliptic By the rule in verse 58, however, it is treated as if 
measured upon a circle of declination, and is, without modification, 
added to or subtracted from the declination, according as the direction 
of the two is the same or different . The commentary takes note, of this 
error, but explains it,; as:, in other similar cases, as being, "for .fear of 

G" 'ng inen H ro nbl e , a n d t m acco u nt x>f the ve^tyfstight inaccliracy, ovjr- 
:e»b3fcilfc«lBfeMs^Sutf,,m^ : > '*: ;••* 

x^jKteeirt inetfce antfelw^bteth* i&ufts ofti* ft&k^*m&lti& 
teti^giWeeJsditnde^tbe de4lJ^o^a^4^e^«#^^eWna«on *&*&<&& 
feYobttaA* *f 'jdfrt^plafeeteFat: *e^^^^wlw%^tn«ir l(^t«^¥a¥ 
afea4^i^n;fett^ Ttio dedtaatfoa fretfetftted bySthVMfe _ra;vers£ 
28 of this" chapter, the precessiott'at'tfce ~^veh time 'being, w ftuisT 
under verses 9-12 of the next chapter, 20° 24' 39". Upon the line for 
tfce sun in ^he table are given the results of the process for calculating 
his declination, the equinor itself being accounted as a " node" ; itis, in 
facts styled, in modern Hindu astronomy, Jcrantipata, "node of declina- 
tion," iltbough that term does not occur m this treatise. 



Results of 


the Process for finding the Latitude and Declination of the 
Planets. 


y Five* 


. Longitude 
' Of Node. 


4o. 
corrected. 


Distance 

from Node. 


Sine. 


Latitude. 


Declination. 


Oeereoted 
Declination. 


Stan, 
Moon, 
Mercury, ' 

^€tlttS, 

itars, 

Jupiter, 

Saturn, 


o 20 24 38 
9 a4 a4 43 
o 20 4o 4t 
i 29 39 22 
r 10 3 5 

2 19 40 5 

3 10 20 45 


• • 

22 43 

1 29 16 

2 i3 47 

2 22 45 

3 14 38 


• • 

9 8 4o 
I 23 14 

3 24 i4 
8 22 34 

4 4 58 
it 26 
t6 24 


3397 
2 7 54 
3i34 
3.409 
2816 
682 
970 


3 36 N. 
2 4N. 
1 21 S. 
1 4N. 
i5N. 
37 N. 


• 1 

23 4i S. 

4 56 N. 

23 10 S. 

20 27 S. 
14 52 8. 

21 42 N. 
i4 4oN 


B 3a ri. 

21 6 *". 
21 48 ?. 
i3 48 *. 
21 57 N. 
fSi7K 



We are now able to compare the Hindu determinations of the ttntf" 
places and motions of the pJaneti-wWr their actaial' position* and ttottattLj 



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ii. 59.] 



S&rya-Siddhdnta. 



231 



as obtained by modem science. The comparison is made in the annexed 
table. As the longitudes given by the Surya-Siddhanta contain a con- 
stant error of 2° 20', owing to the incorrect rate of precession adopted 
by the treatise, and the false position thence assigned to the equinox, we 
give, under the head of longitude, the distance of each planet both from 
the Hindu equinox, and from the true vernal equinox of Jan. 1, 1860. 
The Hindu daily motions are reduced from longitude to right ascension 
by the rule given in the next following; verse (v. 69). The modern data 
are' taken from the American Nautical Almanac. 

True Places and Motions of the Planets, Jan. 1st, 1860, midnight, at 
Washington, according to the Surya-Siddhanta and to Modern Science. 





True Longitude. 




Daily Motion 


Planet 


Surya Siddh&ota : 




Declination. 


in Right Ascension. 




from 
Hindu eq. 


from 
true eq. 


Moderns. 


Surya- 
Siddhanta 


Moderns. 


Sfirya- 
SitWhanta. 


Modems. 


Son, 


278 4o 


276 20 


280 5 


23 4i S. 


23 5S. 


+ 66 a 


+ 66 r8 


Mood, 


8 4 


5 44 


7 37 


8 3a N. 


6 56 N. 


+683 5o 


+655 14 


Mercury, 


255 16 


252 56 


257 25 


21 6S. 


20 42 $, 


+ 3i i3 


+ 5a 39 


Ven as, 


3o5 


3oa 4o 


3o3 2$ 


21 48 S. 


20 58 S. 


+ 7a 59 


+ 78 6 


Man, 


219 10 


si* 5» 


221 33 


i3 46 S. 


14 23 6. 


+ 3i 58 


+ 36 19 


Jupiter, 


1 14 36 


112 16 


111 34 


21 57 N. 


22 iN. 


- 8 at 


- 8 17 


Saturn, 


i4i 27 


»3p 7 


i45 32 


i5 17 N. 


14 i5N. 


- 3 3 


- a 29 



The proper subject of the second chapter, Ilia determination of the 
true places of the planets, bailiff thus brought to a close, we should ex- 
pect to tee the chapter eoncjeded here, and the other matters which it 
contains pat off to that which Mows, in which they would aeem more 
properly to belong. The treatise, however, ie nowhere distingwshed Jbe 
ita orderly end consistent arrangement 

59. Multiply the daily motion of a planet by the time of 
rising of the sign in which it is, and divide by eighteen hun- 
dred ; the quotient add to, or subtract from, the number of respi- 
rations in a revolution : the result is the number of respirations 
in the day and night of that planet 

In the first half of this verse is taught the method of finding the in- 
crement or decrement of right ascension corresponding to the increment 
or decrement of longitude made by any planet during one day. For the 
"time of rising" (udayapr&nas, or, more commonly, uday&xwes, liter- 
ally "respirations of rising") of the different signs, or the time in respi- 
rations (see i. 11), occupied by the successive signs of the ecliptic in 
passing the meridian — or, at the equator, in rising above the horison— 
see verses 42-44 of the next chanter. The statement upon which the 
rule is founded is as follows : if the given sign, containing 1800 / of alt 
(each minute of arc corresponding, as remarked above, under i. 11-19) 
to a respiration of sidereal time), occupies the stated number of respira- 
tions in passing the meridian, what number of respirations will be occu- 
pied by the arc traversed by the planet en a given day! The result 
gives, the .amount hy which the day of each planet, reckoned in the 



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282 



E. Burgess, etc., 



[ii. 59- 



manner of tbis Siddh&nta, or from transit to transit across the inferior 
meridian, differs from a sidereal day : the difference is additive when the 
motion of the planet is direct ; subtractive, when this is retrograde. 

Thus, to find the length of the son's day, or the interval between two 
successive apparent transits, at the time for which his true longitude and 
rate of motion have already been ascertained. The sun's longitude, as 
corrected by the precession, is 9 s 8° 40' ; he is accordingly in the tenth 
sign, of which the time of rising (uday&sava*), or the equivalent in right 
ascension, is 1935 p . His rate of daily motion in longitude is 61' 26". 
Hence the proportion 

1800' : io35P : : 6i' 26" : 66P.04 
shows that his day differs from the true sidereal day by ll v 0P.04. As 
his motion is direct, the difference is additive : the length of the appar- 
ent day is therefore 60 n ll v 0P.04, which is equivalent to 24 h m 27 8 .5, 
mean solar time. According to the Nautical Almanac, it is 24 h m 28 B .6. 
By a similar process, the length of Jupiter's day at the same time is 
found to be 59* 58* 4P, or 23& 65 m 30«.8 ; by the Nautical Almanac, it 
is 23 h 55 m 30 B . 

60. Calculate the sine and versed sine of declination : then 
radius, diminished by the versed-sine, is the day-radius: it is 
either south or north. 

The quantities made use of, and the processes prescribed, in this and 
the following verses, may be explained and illustrated by means of the 
annexed figure {Fig. 8). 

Let the circle ZSZ'N represent the meridian of a given place, C 



Fig. 8. 




in Terse 60 is called the "day-radius.*' 



being the centre, the 
place of the observer, 
S N the section of the 
plane of his horizon — 
S being the south, and 
Nthe north point — Z 
and Z' the zenith and 
its opposite point, 
t!ie nadir, P and P' 
the north and south 
poles, and E and E' 
the points on the me- 
ridian cut by the 
equator. Let E D be 
the declination of a 
planet atagiven time ; 
then D D' will be the 
diameter of the circle 
of diurnal revolution 
described by the 
planet, and BD the 
radius of that circle : 
B D is the line which 
Draw DF perpendicular to EC : 



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ii 63.] S&rya-JSSddhdtita. 2£8 

then it is evident that BD is equal to EC diminished bj EF, which is 
the versed sine of ED, the dedication. 

For " radius" we have hitherto had only the term trijyd lor its equiv- 
alents, trijivd, tribhvjivd, tribhojyd^ tribkamAumrikd), literally " the sins 
of three signs," that is, of 90°. That term, however, is applicable only 
to the radius of a great circle, or to tabular radius. Jn this verse, 
accordingly, we have for "day-radius" the word dinavyAsadaJa, "hall- 
diameter of the day ;" and other expressions synonymous with this am 
found used instead of it in other passages. A luore frequent name for 
the same quantity in modern Hindu astronomy is dyvjyd, "day-sine:?' 
this, although employed by the commentary, is not found anywhere m 
our text 

It is a matter for surprise that we do not find the day-radius declared 
equal simply to the cosine (kotijyd) of declination. 

In illustration of the rule, it will be sufficient to find the radius of {he 
diurnal circle described by the sun at the time for which his place has 
been determined. His declination, E d (Fig. 8) was found to be 23* 41': 
of this the versed sine, EF, is, by the table given above (ii. 22-27), 
200': the difference between this and radius, EC, or 3438 , is 3148, 
.which ia the value of C F or bd r the day-radius. . The declination in 
this case being south* the day-radius is also south of Ata . equator* . ~ 

61. Multiply the sine of decimation by the eauinoctiat shadow, 
£&<$ divide l>y twelve ; the result is the eartii-aiae (kthitifyd}; 
^is,- multiplied 1 ^radius and tJtYSded-bythe d&Y-fiadfus, &iro 
the sine of the ascensional diflTerenoe {corajs tW-humfc& ^of 

^resptrattons due to the ascensional'difiterenoe' 1 " ' : ^ *-' •*- 

62. R shown by the corresponding arc. Add these to, and 
subtract them from, the fourth part of thf corresponding day 
and night, and the sum and remainder axe, when declination is 
north, the half-day and half-night; 

63. When declination is south, the reverse ; these, multiplied 
by two, are tfce day and the night. The day and the night of 
the asterisms (bka) may be found in like manner, by means of 
their declination, increased or diminished by their latitude. 

We were taught in verse 59 how to find the length of the entire day 
Of a planet at any given time ; this passage gives us the method of 
ascertaining the length of its day and of its night, or of that part of the 
day during which the planet is above, and that during which it is below, 
the horizon. 

In order to this, it is necessary to ascertain, for the planet in question, 
its ascensional difference (carer), or the difference between its right and 
oblique ascension, the amount of which varies with the declination of 
the planet and the latitude of the observer. The method of doing this 
is stated in verse 61 : it may be explained by means of the last figure 
(Fig, 8). First, the value of the line AB, which is called the "earth- 
sine" (kshitijyd), is found, by comparing the two triangles ABC and 
C H E, which arc similar, since . the angles AC B and C E II are each 
equal to the latitude of *he observer. The triangle f HE is Represented 
vol. vi. 30 



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284 B. Burgess, etc., pi. 63- 

here by a triangle of which a gnomon of twelve digit* ia the perpen- 
dicular, and Ha equinoctial ehadow, east whet the warn is In the equator 
and on the meridian (seethe neit ehapter, verse 7, etc.), is the base* 
Hence the proportion EH : H C : : BC : AB is equivalent — aince BO 
equals D F, the sine of declination — to gnom. : eq. shad, : : sin decl. : 
earth-sine. But the are of which AB is sine is the same part of the 
circle of diurnal revolution as the ascensional difference is of the equa- 
tor; hence the redaction of A B to the dimensions of a great circle, by 
the proportion BD:AB::CE:CG, gives thevahie of CO, the sine 
of the ascensional difference. The corresponding arc is the measure in 
time of the amount by which the part of the dinrnal circle intercepted 
between the meridian and the horizon differs from a quadrant, or by 
which the time between sun-rise or sun-set and noon or midnight differs 
from a quarter of the day. 

In illustration of the process, we will calculate the respective length 
of the sun's day and night at Washington at the time for which our 
previous calculations have been made. 

. The latitude of Washington being 3Q° 54 f , the length of the equi- 
noctial shadow cast there by a gnomon twelve digits long is found, by 
the rule given below (iii. 17), to be 9 d ,68. The sine, d¥ or b C, of the 
sun's declination at the given time, 23° 41' S, is 1380'. Hence the 
proportion 

ia :9.6ft:: i38o: ui3 

S'ves us the value of the earth-sine, a b, as 1113'. This is reduced to 
e dimensions of a great circle by the proportion 
3i48:3438::iu3:i3i6 
The value of Cg 9 the sine of ascensional difference, is therefore 1216'; 
the corresponding arc is 20° 44', or 1244', which, as a minute of arc 
equals a respiration of time, is equivalent to 3 n 27 v 2p. The total 
length of the day was found above (under v. 59) to be 60 n ll v ; in- 
crease and diminish the quarter of this by the ascensional difference, 
and double the sum and remainder, and the length of the night is found 
to be 37 D V IP, and that of the day 23 n 10 v 5P, which are equivalent 
respectively to 14* 1 45 ln 38 B .6 and 9* 14 m 48 9 .9, mean solar time. . 

Of course, the respective parts of a sidereal day during which each 
of the lunar mansions, as represented by its principal star, will remain 
above and below the horiaon of a given latitude, may be found in the 
same manner, if the declination of the star is known ; and this is stated 
in the chapter (ch. viii) which treats of the asterkms. 
. Whv, A Bis called ksMtijyA is not easy to see* One is tempted 
to understand the term aa meaning rather "sine of situation" than 
"earth-sine," the original signification of hshiti being "abode, reai* 
dence": it might then indicate a sine which, for a given declination, 
varies with the situation of the observer. But that kshiti in this com- 
pound is to betaken in its other acceptation; of "earth," is at least 
strongly indicated by the other and more usual name of the sine in 
question, to/yd, which is used by the eomtnentary, although not in the 
text, and which can only mean " earth-sine." The word cara, used to 
denote the ascensional difference, means simply "variable "; we have 
elsewhere eorukhand*, earadaU, * variable portion" ; that ia to stty r the 



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ii; 65.] SQrya-SiddMnta. 280 

constantly varying amount by which the apparent day and right differ 
from the equatorial day and night of one Jiilfthewhole day each* Tho 
gnomon*' the equinoctial shadow, etc.,: are treated of in the next chapter, 

o4 T The portion (bhoga) of an asterisui (bha) is eight "hundred 
minutes; of a lunar day (titki), in like maimer, seven hundred 
and twenty. If the longitude of a planet, in minutes, be divided 
by the portion of an aaterism, the Jesuit ia it* position in aster* 
isms: by means of the daily motion are found the days, etc . 

The ecliptic is divided (see ch. viii) into 21 lunar mansions Or aster- 
isms, of equal amount ; hence the portion of the ecliptic occupied by- 
each asterism is' 13° 20^, or 600'. In order to find, accordingly, in 
which asterism, at a given time, the moon or any other of the planets 
is, we have only to reduce its longitude, not corrected by the precession, 
to minutes, and divide by 800 : the quotient is the number of asterisms 
traversed, and the remainder the part traversed of the asterism in whiclh 
the planet is. The last clause of tne verse is very elliptical and o"bscure ; 
according to the commentary, it is to be understood thus : divide by the 
planet's true daily motion the part past, and the part to come of the 
current asterism, and the quotients are tne days and fractions of a day 
which the planet has passed, and is to pass, m that asterism. This in- 
terpretation is supported by the analogy of fhe following verses, and is 
doubtless correct 

The true longitude of the moon was found above (under v. 30) to be 
11« 17° 89', or 20,859'. Dividing by 800, we find* that, at the givetf 
time, the moon is in the 2Vth, or last, asterism, named Revati, of which 
it has traversed 59', and has 741' still to pass over.' Its daily motion 
beinff 737', it has spent 28* 4> t and has yet to continue 1*0* 19* 3P, in 
the'asterism. 

The latter part of this process proceeds upon the assumption that the 
planet's rate of motion remains the same during its whole continuance 
m the asterism. A similar assumption,- it will be noticed, is made in all 
{he processes from verse 59 onward ; its inaccuracy is greatest, of course, 
where the moon's motion is concerned. 

Respecting the lunar day (tithi) see below, under verse 66. 

86. From the number of minutes in the sum of the longitudes 
Of the^sna abd moon are fonnd the yogas*, by dividing that sum 
by the portion (bhoga) of an asterism. Multiply the minutes 
past and to eome of the current yoga by sixty, and divide by 
the sum of the daily motions of the two planets ; the result is 
the time in n&JJs. 

What the yog* i^ is evident from this rule for finding it ;k is the 
periooV of variable length, during which the joint motion in longitude of 
the sun and moon amounts to 13° 2C, the* portion of a lunar roanstoa* 
According to Ckfobreoke (Aft. Res*, uu: 365.? Essays, ii, 362 v 3 63) K tibs 
use of the vogs* is chiefly astjologMjai ? the oceuneence of certain moYa* 
hie festivals is, however, also ?egu]ated by them, and- they a/e so fre- 
quently employed that every Htt^tt ah&*^ 



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286 E. Burges*, etc., pi. 65- 

tying ft* yog* for each day, with the time of it* termination* The 
names of the tw*nty-sev*n yoga* are as follows : 

I. Vishkambha, 10. Gagda. 191 Parigha. 

a. Prtti 11. Vrddhi. ao. ffra. 

3. Ayushmank 1a. Dhrura. 31. Siddha. 

4. Saubbagya. i3. Vyaghata. aa. Sadhya. 

5. Qobhana. ii Harahaua. a3. £ubha. 

6. Atigaoda. i5. Vajra. a4. $utfa. 

7. Sukarmao. 16. SiddhL a5. Brahman. 

8. Dhrti. 17. Vyatlpata. a6. Indra. 

91 gala. 18. Varfyas. a 7 . VaidhrtL 

There is also in use in India (see Colebrooke, as above) another 
system of yogas, twenty-eight in number, having for the most part 
different names from these, and governed by other rules in their succes- 
• sion. Of this system the Surya-Siddhanta presents no trace. 

"We will find the time in yogas corresponding to that for which the 
previous calculations have been made. 

The longitude of the moon at that time is 11 s 17° 39', that of the 
sun is 8« 18° 16'; their sum is 8« 6° 54', or 14,754'. Dividing by 800, 
we find that eighteen yogas of the series are past, and that the current 
one is the nineteenth, Parigha, of which 354' are past, and 440' to 
come. To ascertain the time at which the current yoga began and that 
at which it is to end, we divide these parts respectively by 798'£, the 
sum of the daily motions of the sun and moon at the given time, and 
multiply by 60 to reduce the results to nadls : and we find that Parigha 
begin 26 n 36 T before, and will end 33" 30* 4P after the given time. 

The name yoyo, by which this astrological period is called, is applied 
to it, apparently, as designating the period during which the "anm" 
(yoga) of the increments in longitude of the sun and moon amounts to a 

S'ven quantity. It seems an entirely arbitrary device of the astrologers, 
ting neither a natural period nor a subdivision of one, not being of 
any use that we can discover in determining the relative position, or 
aspect, of the two planets with which it deals, nor having any assignable 
relation to the asterisms, with which it is attempted to be brought into 
connection. Were there thirty yogas, instead of twenty-seven, the 
period would seem an artificial counterpart to the lunar day, which is 
the subject of the next verse ; being derived from the sum, as the other 
from the difference, of the longitudes of the sun and moon. 

66. From the number of minutes in the longitude of the moon 
diminished by that of the sun are found the lunar days (tithx), 
by dividing the difference by the portion (bhoga) of a lunar daj. 
Multiply the minutes past and to come of the current lunar day 
by sixty, and divide by the difference of the daily motions of 
the two planets : the result is the time in nfidfa. 

The tithij or lunar day, is (see i, 13) one thirtieth of a lunar month, 
or of the time during which the moon gains in longitude upon the sun 
a whole revolution, or 860°: it is, therefore, the period during which 
the difference of the increment of longitude of the two planets amounts 



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iL6fc] Saryh-SuMli&nta. 287 

to 12°, or 1W, which arc, as stated in verse 64, is its portion (bhoga). 
To find the current lunar day, we divide by this amount the whole ex- 
cess of the longitude of the moon over tnat of the sun at the given 
time ; and to find the part past and to come of the current day, we con- 
vert longitude into time in a manner analogous to that employed in the 
case of the yoga. 

Thus, to find the date in lunar time of the midnight preceding the 
first of January, 1860, we first deduct the longitude of the sun from that 
of the moon ; the remainder is 2 s 29° 24', or 5364' : dividing by 720, 
it appears that the current lunar day is the eighth, and that 324' of its 
portion are traversed, leaving 396' to be traversed. Multiplying these 
numbers respectively by 60, and dividing by 675' 38", the difference of 
" the daily motions at the time, we find that 28* 46 v 2P have passed since 
the beginning of the lunar day, and that it still has 35* 10 v 8P to rim. 

Hie lunar days have, for the most part, no distinctive names, but 
those of each half month (paksha — see above, under i. 48-51) are 
called first, second, third, fourth, etc, up to fourteenth. The last, or 
fifteenth, of each half has, however,, a special appellation : that which 
concludes the first, the light half, ending at the moment of opposition, 
is called paurnamast, p&rmmd, pbrnamd, "day of full moon;" that 
which closes the month, and ends with the conjunction of the two 
planets, is styled amdvdsyA, "the day of dwelling together." 

Each lunar day is farther divided into two halves, called karana, as 
appears from the next following passage. 

67. The fixed (dhruva) karanas, namely fahuni, ndga, oatueh- 
pada the third, and kinstughna, are counted from the latter half 
of the fourteenth day of die dark kalf-month. 

68. After these, the karanas called movable (cam), namely 
bava, etc., seven of them : each of these karapas occurs eight 
times in a month. 

69. Half the portion (bhoga) of a lunar day is established as 
that of the karagjus .... 

Of the eleven karanas, four occur only once in the lunar "mouth, 
while the other seven are repeated each of them eight times to fill out 
the remainder of the month. Their names, and the numbers of the 
half hmar days to which each is applied, are presented below : 

i. Einstqghna. lit 

and, oth, 16th, a3rd, 36th, 37th, 44th, 5»t 

3rd, 10th, 17th, a4th, 3ist, 38th, 45th, 5and 

4th, nth, 18th, a5th, 3and, 3oth, 46th, 53rd. 

5th, lath, 19th, 26th, 33rd, 4oth, 47th, 54th. 

6th, iSth, 20th, 27th, 34th, 4ist» 48th, 55th. 

7th, i4th, aiat, 28th, 35th, 4and, 4otb, 56th. 

8th, iSth, 22nd, 29th, 36th, 43rd, 5oth, 57th. 

58th. 

5 9 th. 

60th. # 



2. Bara. 


3. Balara. 


4. Kaulara. 

5. TAitikL 


6. Gara. 


7. Banij. 

8. Viahti. 


9. Qakuni. 
ia Kftga. 
11. Catushpada. 



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2S8 E. Burgess, etc., [ii. 69- 

Most of these names are very obscure: die last three mean "hawk," 
" serpent," and " quadruped." Karana itself is, by derivation, M factor* 
cause:" in what sense it is applied to denote these divisions of the 
month, we do not know. Nor have we found anywhere an explanation 
of the value and use of the karanas in Hindu astronomy or astrology. 

The time which we have had in view in our other calculations being, 
as is shown under the preceding passage, in the first half of the eighth 
lunar day, is, of course, in the fifteenth karana, which is named Yishti. 

The remaining half-verse is simply a winding-up of the chapter. 

69 Thus has been declared the corrected (sphuta) mo- 
tion of the sun and the other planets. 

The following chapter is styled the "chapter of the three inquiries n 
(triprafnAdhik&ra). According to the commentary, this means that it 
is intended by the teacher as a reply to his pupil's inquiries respecting 
the three subjects of direction (dip), place (depa), and time {k&la). 



CHAPTER III. 



OF DIRECTION, PLACE, AND TIME, 

OoNTKnw:— 1-6, construction of the dial, and description of its- parts; 7, the 
measure of amplitude ; 8, of the gnomon, hypothenuse, and shadow, any two 
being given, to find the third* 9-12, precession of (he equinoxes; 12-18, the 
equinoctial shadow ; 1 8-14, to find, from the equinoctial shadow, the latitude and 
co-latitude; 14-17, the sun's decimation being known, to find, from a given 
shadow at noon, his senith -distance, the latitude, and its sine and cosine; 17, lati- 
tude being given, to find the equinoctial shadow; 17-20, to find, from the lati- 
tude and the sun's zenith-distance at noon, his declination and his true and mean 
longitude; 20-22, latitude and declination being given, to find the noon-shadow 
and hypothenuse ; 22-23, from the sun's declination and the equinoctial shadow, 
to find the measure of amplitude ; 28-25, to find, from the equinoctial shadow 
and the measure of amplitude at any given time, the base of the shadow ; 25-27, 
to find the hypothenuee of the shadow when the sun is upon the prime vertical ; 
27-28, the sun's declination and the latitude being given, to find the sine and the 
measure of amplitude ; 26-88, to find the sines of the altitude and aenhh-distanee 
of the sun, when upon the south-east and south-west vertical circles; 88-84, to 
find the corresponding shadow and hypothenuee; 84-86, the sun's ascensional dif- 
ference and the hour-angle being given, to find the sines of his altitude and senith- 
distance, and the corresponding shadow and hypothennse ; 87-39, to find, by a 
contrary process, from the shadow of a given time, the sun's altitude and senith- 
distance, and the hour-angle ; 40-41, the latitude and the sun's amplitude being 
known, to find his declination and true longitude ; 41-42, to draw the path de- 
scribed by the extremity of the shadow ; 42-46, to find the arcs of right and 
oblique ascension corresponding to the several signs of the ecliptic; 46-48, the 
son's longitude and the time being known, to find the point of the ecliptic which 
is upon the horizon ; 49, the sun's longitude and the how-angle being known, to 
find the point of the eJIptic which is upon the meridian ; 50-51, determmatioa 
of time by means of these data. 



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iit 5.] SdryarSiddhdnia. 239 

1. On a stony surface, made water-level, or upon hard plaster, 
made level, there draw an even circle, of a radius equal to any 
required number of the digits (angula) of the gnomon (ganku). 

2. At its centre set up the gnomon, of twelve digits of the 
measure fixed upon ; and where the extremity of its shadow 
touches the circle in the former and, after parts of the day, 

3. There fixing two points upon the circle, and calling them 
the forenoon and afternoon points, draw midway between them, 
by means of a fish-figure (timi), a north and south line. 

1 4. Midway between the north and south directions draw, by 
a fish-figure, an east and west line : and in like manner also, by 
fish-figures (matsya) between the four cardinal directions, draw 
the intermediate directions. 

5. Draw a circumscribing square, by means of the lines going 
out from the centre; by the digits of its base-line (bhigwkura) 
projected upon that is any given shadow reckoned. 

In this passage is described the method of construction of the Hindu 
dial, if that can properly be called a dial which is without hour-lines, 
and does not give the time by simple inspection. It is, as will be at 
once remarked, a horizontal dial of the simplest character, with a verti- 
cal gnomon. This gnomon, whatever may be the length chosen for it, 
is regarded as divided into twelve equal parts called digits (angula, 
* finger"). The ordinary digit is one twelfth of a span (vitasti), or one 
twenty-fourth of a cubit (hasta) : if made according to this measure, 
then, the gnomon would be about nine inches long. Doubtless the first 
gnomons were of such a length, and the rules of the gnomonic science 
were constructed accordingly, "twelve" and "the gnomon" being used, 
as they are used everywhere in this treatise, as convertible terms : thus 
twelve digits became the unvarying conventional length of the Btaff, and 
all measurements of the shadow and its hypothenuse were made to cor- 
respond. How the digit was subdivided, we have nowhere any hint. 
In determining the directions, the same method was employed which is 
still in use ; namely, that of marking the points at which the extremity 
of the shadow, before and after noon, crosses a circle described about 
the base of the gnomon ; these points being, if we suppose the sun's 
declination to have remained the same during the interval, at an equal 
distance upon either side from the meridian line. In order to bisect the 
line joining these points by another at right angles to it, which will be 
the meridian, the Hindus draw the figure which is called here the " fish" 
{mat*ya or tvmi) ; that is to say, from the two extremities of the line in 
question as centres, and with a radius equal to the line itself arcs of 
circles are described, cutting one another in two points. The lenticular 
figure formed by the two arcs is the "fish;" through the points of 
intersection, which are called (in the commentary) the "mouth" and 
'<* tail," a line is drawn, which is the one required. The meridian being 
thus determined, the east and west line, and those for the intermediate 
points of directions, are laid down from it, by a repetition of the same 
pKXttsSji . A square (caturasra, " having four corners") is then farther 



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240 JB. Burgess, ete: } [ili. 5- 

described about the general centre, or about a circle drawn about that 
centre, the eastern and western sides of which are divided into digits; 
its nse is, to aid in ascertaining the u base" (bhuja) of any given shadow, 
which is the value of the latter when projected upon a north and south 
line (see below, vv. 28-25) ; the square is drawn, as explained by the 
commentary, in order to insure the correctness of the projection, by a 
line strictly parallel to the east and west line. 

: The figure (Fig. 0) given below, under verse 1 r will illustrate the form 
6f the Hindu dial, as described in this passage. 

The term used for "-gnomon" is faitftu, which means simply u staff." 
For the shadow, we-have the common word ch&yd, "ahadowf ' and also, 
in many places, pmbhb and -bhd, which properly signify the. very oppo- 
site of shadow, namely ** light, radiance r it is difficult to see how. they 
should come to be- used in this- sense; ~sd~fttr«s we one aware,, they are 
applied to no other shadow &an that- of- the gnomon. - . 

I 6. The east and .west line is called the* prime vertical (tamd- 
mmdala); it is likewise* denominated the east and west hour 
circle (unma^ufala) and the equinoctial circle (vishumnmandd**)- 

* The line drawn east and west through the base of the gnomon may 
be regarded as the line of common intersection at that point of three 
great circles, as being a diameter to each of the three, and as thus enti- 
tled to represent them all. These circles are the ones which in the last 
Jtgure (Fig. 8, p. 282) are shown projected in their diameters Z Z', P P*, 
and E E' ; the centre 0, in which the diameters intersect, is itself the 
projection of the line in question here. ZZ' represents the prime verti- 
cal, which is styled samamandala, literally " even circle :" P P* is the 
hour circle, or circle of declination, which passes through the east and 
West points of the observer's horizon; it is called unmandala, "up-ciiv 
cle " — that is to say, the circle which in the oblique sphere is elevated ; 
E E' finally, the equator, has the name of vishuvanmandala, or vishuvad- 
vrtta, "circle of the equinoxes;" the equinoctial points themselves 
being denominated vishuvat, or vishuva, which may be rendered u point 
of equal separation." The same line of the dial might be regarded as 
the representative in like manner of a fourth circle, that of the horiaon 

Jkshitija), projected, in the figure, in S N : hence the commentary adds 
b also to the other three ; it is omitted in the text, perhaps, because it is 
represented by the whole circle drawn, about the base of the gnomon, 
and not by this diameter alone. 

. The specifications of this vette, especially of the latter Tialf of it, are 
of little practical importance in the treatise, for there' hardly arises a 
case, in any of its calculations, in which the east and west axis of the 
dial comes to be taken as standing for these circles, or any one of them. 
In drawing the base (bhuja) of the shadow, indeed, it dees represent the 
plane of the prime vertical (see below, under vv. 23-25) ; but this is 
not distinctly stated, and the name of the prime vertical (samamandala) 
occurs in only one other passage (below, v. 26) : the east and west hour- 
circle (unmandala) is nowhere referred to again : and the equator, as 
will be seen under the next verse, is properly represented on the dial, 
not by its east and west axis, but by the line of the equinoctial shadow. 



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iiL 1.] 



S&rya-Siddhdnta. 



241 



7. Draw likewise an east and west line through the extremity 
of the equinoctial shadow (vishuvadbhd) ; the interval between 
any given shadow and the line of the equinoctial shadow is de- 
nominated the measure of amplitude (agrd). 

The equinoctial shadow is defined in a subsequent passage (vv. 12, 
13) ; it is, as we have already had occasion to notice (under ii. 61-63), 
the shadow cast at mid-day when the sun is at either equinox — that if 
to say, when he is in the plane of the equator. Now as the equator is 
a circle of diurnal revolution, the line. of intersection of its plane with 
that of the horizon will be an east and west line ; and since it is also a 
great circle, that line will pass through the centre, the place of the ob- 
server: i£ therefore, we draw through the extremity of the equinoctial 
shadow a line parallel to the east and west axis of the dial, it will repre- 
sent the intersection with the dial of an equinoctial plane passing 
through the top of the gnomon, and in it will terminate the lines drawn 
through that point from any point in the plane of the equator ; and 
hence, it will also coincide with the path of the extremity of the shadow 
on the day of the equinox. Thus, let the following figure (Fig. 9) rep- 
resent the plane of the dial, N 3 and £ W being its two axes, and b the 
base of the gnomon : and let the shadow cast at noon when the sun is 
upon the equator be, 

in a given latitude, '£' ' 

be: then be is the 
equinoctial shadow, 
and Q Q', drawn 
through e and paral- 
lel to EW, is the 
path of the equinoc- 
tial shadow, being 
the line in which a 
ray of the sun, from 
any point in the plane 
of the equator, pass- 
ing through the top 
of the gnomon, will 
meet the face of the 
dial. In the figure 
as given, the circle 
is supposed to be described about the base of the gnomon with a radius 
of forty digits, and the graduation of the eastern and western sides of 
the circumscribing square, used in measuring the base (bhvja) of the 
shadow, is indicated : the length given to the equinoctial shadow corre- 
sponds to that which it has in the latitude of Washington. 

It is not, however, on account of the coincidence of Q Q' with the 
path of the equinoctial shadow that it is directed to be permanently 
drawn upon the dial-face : its use is to determine for any given shadow 
its o^rd, or measure of amplitude. Thus, let bd, bd', b *, 6/, bm f be 
shadows cast by the gnomon, under various conditions of time and dec- 
lination : then the distance from the extremity of each of them to the 

VOL. VI. 31 




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242 M Burgess, etc., [iii. f- 

line of the equinoctial shadow, or de, d'e y ktf, le", m-e 1 " respectively, is 
denominated the agr& of that shadow or of that time. 
- The term aprd we have translated " measure of amplitude," because 
it does in fact represent the sine of the sun's amplitude—understanding 
by M amplitude" the distance of the sun at rising or setting from the 
east or west point of the horizon — varying with the hypothenuse of the 
shadow, .and always maintaining to that hypothenuse the fixed ratio of 
the. sine of amplitude to radius. That this is so, is assumed by the text 
in its treatment of the agrb, but is nowhere distinctly stated, nor is the 
commentator at the pains of demonstrating the principle. Since, how* 
aver, it is not an immediately obvious one, we will take the liberty of 
giving the proof of it .-.v. J • 

In the annexed figure (Fig. 10) let C represent the top of the gnomon, 
and let K be any given position of the sun in the heavens. From K 
lira w KB' at right angles to the plane of the prime vertical, meeting that 

Fig. 10. 




jfdabe in TV, and let the point of ite intersection with the plane of the 
iquatorbe in P. Join KC, EXC, and B'C. Then KC is radius, arid 
£'K is eonal to the aine of the sun's amplitude: for if, in the son's 
daily revolution, the point K is brought to the horizon, E' B' will disap- 
pear, KE'C will become a right angle, KCE' will be the amplitude, 
and P K its sine ; but, with a given declination, the value of E'K re- 
mains always the same, since it is a line drawn in a constant direction 
between two parallel planes, that of the circle of declination and that of 
the equator. Now conceive the three lines intersecting in to be pro* 
duced until they meet the plane of the dial in A*, «*, and k respectively; 
these three points will be in the same straight line, being in the line of 
intersection with the horizon of the plane K B f C produced, and this 
line, b'k, will be at right angles to B'ft', since it is the line of intersec- 
tion of two planes, ^ach of which is at right angles to the plane of the 
prime vertical, in which B* 1/ lies. K B' and k V are therefore parallel, 
and the triangles C E' K and C *h are similar, and * k : C k : : E' K : C K. 
But C£ is the hypothenuse of the shadow at the given time, and e?k is 
flie *<?*&, or measure of amplitude, since «f, by what was said above, is 
in the line of the equinoctial shadow; therefore meas. ampl.:hyp. 
shad. Train ampUR flenee, if the declination and the latitude, which 
together tletermine the sine of amplitude, be given, the measure of 
amplitude will vary with the hypothenuse of the shadow, and the 



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iii. 10.] S&rya-Siddh&nta. 248 

measure of amplitude of any given shadow will be to that of any other, 
as the hypothennse of the former to that of the latter. 

The lettering of the above figure is made to correspond, as nearly as 
may be, with that of the one preceding, and also with that of the one 
given later, under verses 13 and 14, in either of which the relations of 
the problem may be farther examined. 

- There are other methods of proving the constancy of the-tatie borne 
by the measure of amplitude to the- hypothenuse of the shadow, tat we 
have chosen to-give the* one which: seemed to us -ttbst likely- to her that 
by- which the Hindus themselves deduced it. Our demonstration-is 4n 
one. respect only- liable to objection- as representrtg a Hindu process-*- 
it is founded, namely, upon the comparison of obHqee-angied triangle^ 
which elsewhere in this treatise are hardly employed at all. mill, 
although the Hindus hA no methods -of solving problems excepting 
in right-angled trigonometry, it is hardly to be- supposed that- they re- 
frained from deriving proportions from the similarity of oblique-angled 
triangles. The principle m question admits of being proved by means 
of right-angled triangles alone, but these would be situated in different 
planes. 

Why the line on the dial which thus measures the sun's amplitude » 
called the agrd, we have been unable thus far to discover. The wonj, 
a feminine adjective (belonging, probably, to rtkhd, "line," understood), 
literally means " extreme, first, chief." rossibly it may be in some wajr 
connected with the use of antyd, " final, lowest," to designate the line 
E^ or E G (Fig. 8, p. 232) : see below, under v. 35. The sine of amplify 
fetde itself, a C or AC (Fig. 8), is called below (vv. 27-30) agrajyA. j 

8. The square root of the sura of thesqfcaree of the gnomoi 
and shadow ia the hypothenuse ;. if fipra the square ofrtbe kttttf 
the square of the enqmon be ^btracted,;theaquare root of the 
remainder is the shadow: the .gnomon is found by the.oonvetrste 
process. 

This. is simply an application of the familiar rule,..tb*t m ;^righ£ 
angled triangle the square of the hypothenuse is equal to tljfl. sum of 
the squares of the other two sides^to.the triangle produced, by the* 
gnomon as perpendicular, the shadow, as baseband itie. hypothenuse, ,of 
the shadow, the line drawn from .the top of the gnomon to, the, extrem- 
ity of the shadow, as hypothenuse. 

The subject next considered is. that of the precession of the equinoxes. 

9. In an Age (ywfira), the circle of the astefisms (bhit) faTI^ 
back eastward thirty score of revolutions, Of the result ob* 
tained after multiplying the sum of days (dwtgav&) by this num- 
ber, and. dividing by the,number of natural days in an .Age, [ 
' 10. Take the part which determines the sinj^..ni^tiply it by 
three, and divide dv ten ; thus are found the decrees calbd those 
of the precession (ayana). From the longitude of a planet .is 
corrected by these are to be calculated the declination, shadow 
ascensional difference (caradala) r etc. 



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244 K Burgess, etc., [in. 11- 

« 

11. The circle, as thus corrected, accords with its observed 
place at the solstice (ayana) and at either equinox ; it has moved 
eastward, when the longitude of the sun, as obtained by calcu- 
lation, is less than that derived from the shadow, 

12. By the number of degrees of the difference ; then, turning 
back, it has moved westward by the amount of difference, when 
the calculated longitude is greater. . . . 

Nothing could well be more awkward and confused than this mode of 
stating the important fact of the precession of the equinoxes, of de- 
scribing its method and rate, and of directing how its amount at any 
time is to be found. The theory which the passage, in its present form, 
is actually intended to put forth is as follows : (he vernal equinox librates 
westward and eastward from the fixed point, near f Piscium, assumed as 
the commencement of the sidereal Bphere — the limits of the libratory 
movement being 27° in either direction from that point, and the time 
of a complete revolution of libration being the six-hundredth part of 
the period called the Great Age (see above, under i. 16-17), or 7200 
years ; so that the annual rate of motion of the equinox is 54". We 
will examine with some care the language in which this theory is con- 
veyed, as important results are believed to be deducible from it. 

The first half of verse professes to teach the fundamental fact of the 
motion in precession. The words bhdndm cakram, which we have ren- 
dered ** circle of the asterisms," i. e., the fixed zodiac, would admit of 
being translated u circle of the signs," i. e., the movable zodiac, as 
reckoned from the actual equinox, since bha is used in this treatise in 
either sense. But our interpretation is shown to be the correct one by 
the directions given in verses 11 and 12, which teach that when the sun's 
calculated longitude — which is his distance from the initial point of the 
fixed sphere — is less than that derived from the shadow by the process 
to be taught below (w. 17-19) — which is his distance from the equinox 
— the circle has moved eastward, and the contrary : it is evident, then, 
that the initial point of the sphere is regarded as the movable point, 
and the equinox as the fixed one. Now this is no less strange than in- 
consistent with the usage of the rest of the treatise. Elsewhere t Pis- 
cium is treated as the one established limit, from which all motion com- 
menced at the creation, and by reference to which all motion is reckoned, 
while here it is made secondary to a point of which the position among 
the stars is constantly shifting, and which hardly has higher value than 
a node, as which the Hindu astronomy in general treats it (see p. 230). 
The word used to express the motion (lambate) is the same with that 
employed in a former passage (i. 26) to describe the eastward motion of 
the planets, and derivatives of which (as ktmba, lambanq, etc) are. not 
infrequent in the astronomical language ; it means literally to "lag, hang 
back, fall behind :" here we have it farther combined with the prefix 
pari, * about, round about," which seems plainly to add the idea of a 
complete revolution in the retrograde direction indicated by it, and we 
have translated the line accordingly. This verse, then, contains no. hint 
of a libratory movement, but rather the distinct statement of a contin- 
uous eastward revolution. It should be noticed farther, although the 



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ill 12.] S&rya-Siddh&ifa. 246 

circumstance is one of lees significance, that the form in which the 
number of revolutions is stated, trinfatkrtya*, " thirty twenties," has no 
parallel in the usage of this Siddhanta elsewhere. 

We may also mention in this connection that BhAskara, the great 
Hindu astronomer of the twelfth century, declares in his Siddh&nta- 
^iromani (Gol&dh., vi. 17) that the revolutions of the equinox are given 
by the Surya-Siddh&nta as thirty in an Age (see Colebrooke, As. Res*, 
xii. 209, etc. ; Essays, ii. 374, etc., for a full discussion of this passage 
and its bearings) ; thus not only ignoring the theory of libration* but 
giving a very different nnmber of revolutions from that presented by our 
text. As regards this latter point, however, the change of a single letter 
in the modern reading (substituting trin$atkrtvaa y " thirty times," for 
irinfatkftyas, u thirty twenties") would make it accord with Bh&skara's 
statement. We shall return again to this subject. 

The number of revolutions, of whatever kind they may be, being 600 
in an Age, the position at any given time of the initial point of the 
sphere with reference to the equinox is found by a proportion, as follows: 
as the number of days in an Age is to the number of revolutions in the 
same period, so is the given " sum of days'* (see above, under I 48-51) 
to the revolutions and parts of a revolution accomplished down to the 
given time. Thus, let us find, in illustration of the process, the amount 
of precession on the first of January, 1860. Since the number of years 
elapsed before the beginning of the present Iron Age (kali yuga) is di- 
visible by 7200, it is unnecessary to make our calculation from the com- 
mencement of the present order of things : we may take the sum of 
days since the current Age began, which is (see above, under i 50) 
1,811,945. Hence the proportion ,' 

1,577,917,828* : 6oowr : : 1,81 1,94$* : on* a48° a> 8".o 
gives us the portion accomplished of the current revolution. Of this 
we are now directed (v. 10) to take the part which determines the sine 
(cfo*, or bkvja — for the origin and meaning of the phrase, see above, 
under ii. 29, 30). This direction determines the character of the motion 
as libratory. For a motion of 91°, 92°, 93°, etc., gives, by it, a preces- 
sion of 89°, 88°, 87°, etc. ; so that the movable point virtually returns 
upon its own track, and, after moving 180°, has reverted to its starting- 
place. So its farther motion, from 180° to 270°, gives a precession in- 
creasing from 0° to 90° in the opposite direction ; and this, again, is 
reduced to 0° by the motion from 270° to 360°. It is as if the second 
and third quadrants were folded over upon the first and fourth, so that 
the movable point can never, in any quadrant of its motion, be more 
than 90° distant from the fixed equinox. Thus, in the instance taken, 
the bhuja of 248° 2' 8".9 is its supplement, or 68° 2' 8".9 ; the first 
180° having only brought the movable point back to its original posi- 
tion, its present distance from that position is the excess over 186° 
of the aire obtained as the result of the first process. But this distance 
we are now farther directed to multiply by three and divide by ten : 
this is equivalent to reducing it to the measure of an arc of 27°, instead 
of 90°, as the quadrant of hbration, since 3 : 10 : : 27 : 90. This being 
done, we find the actual distance of the initial point of the sphere from 
the equinox on the first of January, 1860, to be 20° 24' 88".67. 



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246 K Burgess, etc., [iii. 12. 

The question now arises, in which direction is the precession, thos 
ascertained, to be reckoned f And here especially is brought to light 
the awkwardness and insufficiency, and even the inconsistency, of the 
process as taught iu the text Not only hare we no rale given which 
furnishes us the direction, along with the amount, of the processional 
movement, but it would .even be a .fair, and strictly- legitimate deduction 
from, verse 0, that that, movement is. taking place at the present time in 
aji opposite direction from the actual one. We have already remarked 
aWe that the bet complete period of libratory revolution closed with 
tb$ close, of the last Brazen Age, .and the process of calculation -has 
shown that we are now jn.the third quarter x>f- a new period, and in the 
third quadrant .oft the current revolution. Therefore, if the revolution 
is an eastward one, aa taught in the text, only taking place upon a folded 
eircie, so as to.be made libratory,. the present position of the movable 
pointy t Piscium, ought to be to the west of the equinox, instead of to 
the east, *s ft actually is^ It was probably on account of this unfortu- 
nate flaw in .the process, that no rule with, regard to the direction was 
given, excepting the experimental one contained in verses 11 and 12, : 
which, moreover, is. not. properly supplementary to the preceding rules, 
but rather an independent method of determining, from observation, both 
the direction and the amount of the precession. In verse 12, it may be 
remarked, in the word dvrtya, "turning back," is found the only distinct 
intimation to be discovered in the passage of the character of the motion 
as libratory. 

. We have already above (under ii. 28) hinted our suspicions that the 
phenomenon of the precession was made no account of in the original 
composition of the Surya-Siddhanta, and that the notice taken of it b? 
the treatise as it is at present is an afterthought : we will now proceed 
to expose the grounds of those suspicions. 

It is, in the first place, upon record (see Colebrooke, As. Res., xii. 215 ; 
Essays, ii. 380, etc.) that some of the earliest Hindu astronomers were 
ignorant o£ or ignored, the periodical motion of the equinoxes ; Brahma- 
gnpta himself is mentioned among those whose systems took no account 
of it ; it is, then, not at all impossible that the Burya-Siddhanta, if an 
ancient work, may originally have done the same. Among the positive 
evidences to that effect, we would first direct attention to the significant 
fact that, if the verses at the head of this note were expunged, there 
would not be found, in the whole body of the treatise besides, a single 
hint of the precession. Now it is not a little difficult to suppose that a 
phenomenon of so much consequence as this, and which enters as an 
element into so many astronomical- processes, should, had it been borne 
distinctly in mind in the framing of the treatise, have been hidden away 
thus in a pair of verses, and unacknowledged elsewhere — no hint being 
given, in connection with any of the processes taught, as to whether 
the correction for precession is to be applied or not, and only the gen- 
eral directions contained in the latter half of verse 10, and ending with 
an " etcV? being even here presented. It has much more the aspect of 
an after-thought, a correction found necessary at a date subsequent td 
the original composition, and therefore inserted, with orders to "apply 
it wherever it is required," The place where the subject is introduced 



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iii. 12.] S&rya-Siddh&nta. 247 

looks the same way : as having to do with a revolution, as entering into 
the calculation of mean longitudes, it should have found a place where 
such matters are treated o£ in the first chapter ; and even in the second 
chapter, in connection with the rule for finding the declination, it would 
have been better introduced than it is here. Again, in the twelfth 
chapter, where the orbits of the heavenly bodies are given, in terms 
dependent upon their times of revolution, such an orbit is assigned to 
the asterisms (v. 88) as implies a revolution once in sixty years : jfc is, 
indeed, very difficult to see what can have 'been intended by such a revo* 
lution as this ; but if the doctrine respecting* the revolution of th* 
asterisms given in verse 9 of this chapter had been in the mind of the 
author of the twelfth chapter, he would hardly have added -another and 
a conflicting statement respecting the same or a kindred phenomenon. 
It appears to us even to admit of question whether the adoption by the 
Hindus of the sidereal year as the unit of time does not imply a failure 
to recognize the fact that the equinox was variable. We should expect, 
at any rate, that i£ at the outset, the ever-increasing discordance be- 
tween the solar and the sidereal year had been fully taken into account 
by them, they would have more thoroughly established and defined the 
relations of the two, and made the precession a more conspicuous feature 
of their general system than they appear to have done. In the con- 
struction of their cosmical periods they have reckoned by sidereal years 
only, at the same time assuming (as, for instance, above, i. 13, 14) that 
the sidereal year is composed of the two ayanas, "progresses" of the 
sun from solstice to solstice. The supposition of an after-correction 
likewise seems to furnish the most satisfactory explanation of the form 
given to the theory of the precession. The system having been first 
constructed on the assumption of the equality of the tropical and side- 
real years, when it began later to appear, too plainly to be disregarded, 
that the equinox had changed its place, the question was how to intro- 
duce the new element. Now to assign to the equinox a complete revo- 
lution would derange the whole system, acknowledging a different nunv 
ber of solar from sidereal years in the chronological periods ; if, however, 
a libratory motion were assumed, the equilibrium would be maintained, 
since what the solar year lost in one part of the revolution of libration 
it would gain in another, and so the tropical and sidereal years would 
coincide, in number and in limits, in each great period. The circum- 
stance which determined the limit to be assigned to the libration we 
conceive to have been, as suggested by Bentley (Hind. Ast, p. 182), that 
the earliest recorded Hindu year had been made to begin when the sun 
entered the asteri&m Krttika, or was 26° 40' west of the point fixed 

ri as the commencement of the sidereal sphere for all time (see 
e, under i. 27), on which account it was desirable to make the arc 
of libration include the beginning of Krttika. 

Besides these considerations, drawn from the general history of the 
Hindu astronomy, and the position of the element of the precession in' 
the system of the Surya-Siddhanta, we have still to urge tne blind and 
incoherent, as well as unusual, form of statement of the phenomenon/ 
as rally exposed above. There is nothing to compare with it in this 
respect in any other part of the treatise, and we are unwilling to believe 



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248 R Burgess, tic., [iii. 12- 

that in the original composition of the Siddhanta a clearer explanation, 
and one more consistent in its method and language with those of the 
treatise generally, would not have been found for the subject. We even 
discover evidences of more than one revision of the passage. The first 
half of verse 9 so distinctly teaches, if read independently of what follows 
it, a complete revolution of the equinoxes, that, especially when taken in 
connection with Bhaskara's statement, as cited above, it almost amounts 
to proof that the theory put forth in the Surya-Siddhanta was at one 
time that of a complete revolution. The same conclusion is not a little 
strengthened, farther, by the impossibility of deducing from verse 9, 
through the processes prescribed in the following verses, a true expres- 
sion for the direction of the movement at present : we can see no reason 
why, if the whole passage came from the same hand, at the same time, 
this difficulty should not have been avoided ; while it is readily explain- 
able upon the supposition that the libratory theory of verse 10 was 
added as an amendment to the theory of verse 9, while at the same 
time the language of the latter was left as nearly unaltered as possible. 

There seems, accordingly, sufficient ground for suspecting that in the 
Surya-Siddhanta, as originally constituted, no account was taken of the 
precession ; that its recognition is a later interpolation, and was made 
at first in the form of a theory of complete revolution, being afterward 
altered to its present shape. Whether the statement of Bhaskara truly 
represents the earlier theory, as displayed in the Surya-Siddhanta of his 
time, we must leave an undetermined question. The very slow rate 
assigned by it to the movement of the equinox — only 9" a year — throws 
a doubt upon the matter : but it must be borne in mind that, so far as 
we can see, the actual amount of the precession since about A. D. 570 
(see above, under i. 27) might by that first theory have been distributed 
over the whole duration of the present Age, since B. C. 3102. 

In his own astronomy, Bhaskara teaches the complete revolution 
of the equinoxes, giving the number of revolutions in an JEon (of 
4,820,000,000 years) as 199,669 ; this makes the time of a single revo- 
lution to be 21,635.8073 years, and the yearly rate of precession 
59".9007. It is not to be supposed that he considered himself to have 
determined the rate with such exactness as would give precisely the odd 
number of 199,669 revolutions to the JEon; the number doubtless 
stands in some relation which we do not at present comprehend to the 
other elements of his astronomical system. Bhaakara's own commenta- 
tors, however, reject his theory, and hold to that of a libration, which 
has been and is altogether the prevailing doctrine throughout India, and 
seems to have made its way thence into the Arabian, and even into the 
early European astronomy (see Colebrooke, as above). 

Bentley, it may be remarked here, altogether denies (Hind. Ask, p. 
130, etc.) that the libration of the equinoxes is taught in the Surya- 
Siddhanta, maintaining, with arrogant and unbecoming depreciation of 
those who venture to hold a different opinion, that its theory is that of 
a continuous revolution in an epicycle, of which the circumference is 
equal to 108° of the zodiac. In truth, however, Bent-ley's own theory 
derives no color of support from the text of the Siddhanta, and is besides 
in itself utterly untenable. It is not a little strange that he should not 



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iii. 13.] Stirya-Si<Hhdnta. 249 

have perceived that, if the precession were to be explained by a revolu- 
tion in an epicycle, its rate of increase would not be equable, but as the 
increment of the sine of the arc in the epicycle traversed by the mova- 
ble point, farther varied by the varying distance at which it would be 
seen from the centre in different parts of the revolution ; and also that, 
the dimensions of the epicycle being 108°, the amount of precession 
would never come to equal 27°, but would, when greatest, fall short of 
18°, being determined by the radius of the epicycle. BentleyV whole 
treatment of the passage shows a thorough misapprehension of its mean- 
ing and relations: he even eommits the blunder of understanding' the 
first half of verse 9 to refer to the motion of* the equinox, instead of io 
that of the initiah point of the sidereal sphere; - - . . * 

Among the Greek astronomers, Hipparchu« is* regarded as the first 
who discovered the- precession of the equinoxes; their Tate of motion, 
'however, seems not to have been confidently determined by him, 
although he pronounces it to be at any rate not less than 36" yearly. 
For a thorough- discussion of the subject of the precession in Greek 
astronomy see Delambre's History of Ancient Astronomy, ii. 247, etc. 
From the observations reported as the data whence Hipparchus made 
his discovery, Delambre deduces very nearly the true rate of the preces- 
sion. Ptolemy, however, was so unfortunate as to adopt for the true 
rate Hipparchus's minimum, of 36" a year : the subject is treated of by 
him in the seventh book of the Syntaxis. The actual motion of the 
equinox at the present time is 50".25 ; its rate is slowly on the increase, 
having been, at the epoch of the Greek astronomy, somewhat less than 
50". How the Hindus succeeded in arriving at a determination of it so 
much more accurate than was made by the great Greek astronomer, or 
whether it was anything more than a lucky hit on their part, we will 
not attempt here to discuss. 

The term by which the precession is designated in this passage is 
ayan&nfa, " degrees of the ayanaP The latter word is employed in 
different senses : by derivation, it means simply u going, progress," and 
it seems to have been first introduced into the astronomical language to 
designate the half-revolutions of the sun, from solstice to solstice ; these 
being called respectively (see xiv. 9) the uttar&yana and dakskin&yana, 
u northern progress" and u southern progress." From this use the word 
was transferred to denote also the solstices themselves, as we have trans- 
lated it in the first half of verse 11. In the fatter sense we conceive 
it to be employed in the compound ayandnpa ; although why the name 
of the precession should be derived from the solstice we are unable 
clearly to see. Hie term hr&ntipita^ati^ "movement of the node of 
declination," which is often met with in modern works on Hindu astron- 
omy, does not occur in the Surya-Siddhanta. 

12. ... In like manner, the equatorial shadow which is oast 
at mid-day at one's place of observation 

18. Upon the north and south line of the dial — that is the 
equinoctial shadow {viskuvatprdbhd) of that place. ... 

The equinoctial shadow has been already sufficiently explained, in 
connection with a preceding passage (above, v. 7). In this treatise it is 
vol. vi. 32 



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260 B. Burgess, etc., pii. 13- 

known only by names formed by combining one of the words for 
shadow (chdyd, bkd, prabhd), with vishuvat, "equinox" (see above, 
under v. 6). In modern Hindu astronomy it is also called akshabhd, 
"shadow of latitude" — i. e., which determines the latitude — and pala- 
bhd, of which, as used in this sense, the meaning is obscure. 

13 Radius, multiplied respectively by gnomon and shadow, 

and divided by the equinoctial hypothenuse, 

14. Gives the sines of co-latitude (lamba) and of latitude 
(aksha): the corresponding arcs are co-latitude and latitude, 
always south. ... 

The proportions upon which these rules are founded are illustrated 
by the following figure (Pig. 11), in which, as in a previous figure (Fig. 
8, p. 232), Z S represents a quadrant of the meridian, Z being the zenith 
and S the south point, Pig ! j 

being the centre, ^^_|^^p^»^^^^^^^ 
and EC the projec- ■ 
tion of the plane of I 
the equator. In order I W^A 
to illustrate the oor- I 
responding relations I 

of the dial, we have K itffi 

conceived the gno- I 
mon, C b, to be placed IVfl 
at the centre. Then, II 
when the sun is on I 
the meridian and in I 
the equator, at E, the I 
shadow cast, which is I 

the equinoctial shad- ^^ ^^■■■^^^■l^^^^H 

ow, is b *, while C e is the corresponding hypothenuse. But, by simi- 
larity of triangles, 

C«:6*::CE:BE 
and C*:C6::CE:CB 

and as BE is the sine of EZ, which equals the latitude, and CB the 
sine of E S, its complement, the reduction of these proportions to the 
form of equations gives the rules of the text 

14. . . . The mid-day shadow is the base (bhuja) ; if radius be 
multiplied by that, 

15. And the product divided by the corresponding hypothec 
nuse, the result, converted to arc, is the sun's zenith-distance 
(nata), in minutes : this, when the base is south, is north, and 
when the base is north, is south. Of the sun's zenith-distance 
and his declination, in minutes, 

16. Take the sum, when their direction is different — the dif- 
ference, when it is the same; the result is the latitude, in 
minutes. From this find the sine of latitude; subtract its 
square from the square of radius, and the square-root of the re- 
mainder 



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iii. 20.J S&rya-Siddhanta. 251 

17. Is the sine of co-latitude. . . . 

This passage applies to cases in which the sun is not upon the equator, 
but has a certain declination, of which the amount and direction are 
known. Then, from the shadow cast at noon, may be derived his zenith- 
distance when upon the meridian, and the latitude. Thus, supposing 
the sun, having north declination ED (Fig. 11), to be upon the meridian, 
at D : the shadow of the gnomon will be b d 9 and the proportion 

Cd:db::CD:I)B"" 
gives DB"", the sine of the sun's zenith-distance, ZD, which is found 
from it by the conversion of sine into arc by a rule previously given 
(ii. 33). ZD in this case being south, and ED being north, their sum, 
E Z, is the latitude : i£ the declination being south, the sun were at D / , 
the difference of ED 7 and ZI> would be EZ, the latitude* The figure 
does not give an illustration of north zenith-distance, being drawn for 
the latitude of Washington, where that is impossible. The latitude 
being thus ascertained, it is easy to find its sine and cosine : the only 
thing which deserves to be noted in the process is that, to find the co- 
sine from the sine, resort is had to the laborious method of squares, 
instead of taking from the table the sine of the complementary arc, or 
the kotijyd. 

The sun's distance from the zenith when he is upon the meridian is 
called natds, "deflected," an adjective belonging to the noun UpiA$ f M min- 
utes," or bh&g&i, anf&s, " degrees " The same term is also employed, as 
will be Been farther on (vv. 34-36), to designate the hour-angle. For 
zenith-distance off the meridian another term is used (see below, v. 33). 

17. . . . The sine of latitude, multiplied by twelve, and divi- 
ded by the sine of co-latitude, gives the equinoctial shadow. . . . 

That is (Fig. 11), 

BC:BE::Cb:be 
the value of the gnomon in digits being substituted in the rale for the 
gnomon itself! 

17. . . . The difference of the latitude of the place of observa- 
tion and the sun's meridian zenith-distance in degrees (nala- 
bkdg&s), if their direction be the same, or their sum, 

18. If their direction be different, is the surfs declination : if 
the sine of this latter be multiplied by radius and divided by the 
sine of greatest declination, the result, converted to arc, will be 
the sun's longitude, if he is in the quadrant commencing with 
Aries; 

19. If in that commencing with Cancer, subtract from a half- 
circle ; if in that commencing with Libra, add a half-circle ; if in 
that commencing with Capricorn, subtract from a circle : the re- 
sult, in each case, is the true (sphuia) longitude of the sun at 
mid-day. 

20. To this if the equation of the apsis (mdnda phala) be 
repeatedly applied, with a contrary sign, the sun's mean longi- 
tude will be found. . . . 



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262 M Burgess, etc. y [in. 20- 

This passage teaches how, when the latitude of the observer is 
known, the sun's declination, and his true and mean longitudes, may be 
found by observing his zenith-distance at noon. The several parts of 
the process are all of them the converse of processes previously given, 
and require no explanation. To find the sun's declination from his 
meridian zenith-distance and the latitude (reckoned as south, by v. 14), 
the rule given above, in verses 15 and 16, is inverted; the true longitude 
is found from the declination by the inversion of the method taught in 
ii. 28, account being taken of the quadrant in which the sun may be 
according to the principle of ii. 30 : and finally, the mean may be de- 
rived from the true longitude by a method of successive approximation, 
applying in reverse the equation of the centre, as calculated by ii. 39. 

It is hardly necessary to remark that this is a very rough process for 
ascertaining the sun's longitude, and could give, especially in the hands 
of Hindu observers, results only distantly approaching to accuracy. 

20. . . . The sum of the latitude of the place of observation 
and the sun's declination, if their direction is the same, or, in the 
contrary case, their difference, 

21. Is the; sun's meridian zenith-distance (natdngds); of that 
find the base-sine (bahujyd) and the perpendicular-sine (Jcotijyd). 
If, then, the base-sine and radius be multiplied respectively by 
the measure of the gnomon in digits, 

22. And divided Tby the perpendicular-sine, the results are the 
shadow and hypothenuse at mid-day. ... 

The problem here is to determine the length of the shadow which 
will be cast at mid-day when the sun has a given declination, the latitude 
of the observer being also known. First, the sun's meridian zenith-dis- 
tance is found, by a process the converse of that taught in verses 15 and 
16; then, the corresponding sine and cosine having been calculated, a 
simple proportion gives the desired result. Thus, let us suppose the sun 
to be at D' (Fig. 11, p. 250) ; the sum of his south declination, Ely, 
and the north latitude, E Z, gives the zenith-distance, Z D 7 : its sine 
(bkujajyA) is I> B'", and its cosine (kotijyd) is C B w . Then 

CB"':B'"D':":C&:&c*' 
and CB'":CD'::C6:Crf' 

which proportions, reduced to equations, give the value of bd\ the 
shadow, and C d' 7 its hypothenuse. 

22. . . . The sine of declination, multiplied by the equinoctial 
hypothenuse, and divided by the gnomon-sine fankujivd), 

23. Gives, when farther multiplied by the hypothenuse of any 
given shadow, and divided by radius (rnadhyamrTia), the sun's 
measure of amplitude (arhdgrd) corresponding to thatohadow. .' . 

In this passage we are taught* the declination being known, how to 
find the measure of amplitude (agrd) of any given shadow, as prepara- 
tory to determining, by the next following rule, the base (bhuja) of the 
shadow, by calculation instead of measurement. The first step is to find 
the sine of the sun's amplitude : in order to this, we compare the trian- 



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iii. 25.] S&rya-Siddhdnta. 253 

gles ABC and CEII (Fig. 13, p. 254), which are similar, since the 
angles AC B and C E H are each equal to the latitude of the observer. 
Hence EH:EC::BC:AC 

But the triangles CEH (Fig. 13) and Cbe (Fig. 11) are also similar; 
and EH:EC::C6:C« 

Hence, by equality of ratios, C6:C*::BC:AC 
and A C, the sine of the sun's amplitude, equals B C — which is the sine 
of declination, being equal to D F — multiplied by C e, the equinoctial 
hypothenuse, and divided by C 6, the gnomon. The remaining part of 
the process depends upon the principle which we have demonstrated 
above, under verse 7, that the measure of amplitude is to the hypothe- 
nuse of the shadow as the sine of amplitude to radius. 

Why the gnomon is in this passage called the "gnomon-sine," it is 
not easy to discover. Verae 23 presents also a name for radius, madhya- 
karna, " half-diameter," which is not found again ; nor is karna often 
employed in the sense of "diameter" in this treatise. 

23. . . . The sum of the equinoctial shadow and the sun's 
measure of amplitude (arkdgrd), when the sun is in the southern 
hemisphere, is the base, north ; 

24. When. the sun is in the northern hemisphere, the base is 
found, if north, by subtracting the measure of amplitude from 
the equinoctial shadow ; if south, by a contrary process — accord- 
ing to the direction of the interval between the end of the 
shadow and the east and west axis. 

25. The mid-day base is invariably the midday shadow. . . . 

We have already had occasion to notice, in connection with the first 
verses of this chapter, that the base (bkuja) of the shadow is its projec- 
tion upon a north and south line, or the distance of its extremity from 
the east and west axis of the dial. It is that line which, as shown 
above (under v. 7), corresponds to and represents the perpendicular let 
fall from the sun upon the plane of the prime vertical. Thus, if (Fig. 
11, p. 250) K, L, I>, D be different positions of the sun — K and L 
being conceived to be upon the surface of the sphere — the perpendicu- 
lars KB', LB", I^B'", DB"" are represented upon the dial by kb, lb, 
d'b, db, or, in Fig. 9 (p. 241), by kV, lb", d' 6, db. Of these, the two 
latter coincide with their respective shadows, the shadow cast at noon 
being always itself upon a north and south line. The base of any 
shadow may be found by combining its measure of amplitude (at/rd) 
with the equinoctial shadow. When the sun is in the southern hemis- 
phere, as at D' or K (Fig. 11), the measure of amplitude, ed f or ek, is to 
be added always to the equinoctial shadow, b e, in order to give the base, 
b d 1 or b k. If, on the contrary, the sun's declination be north, a differ- 
ent method of procedure will be necessary, according as he is north or 
south from the prime vertical. If he be south, as at V, the shadow, bd, 
will be thrown northward, and the base will be found by subtracting the 
measure of amplitude, de, from the equinoctial shadow, be: if he be 
north, as at L, the extremity of the shadow, I, will be south from the 
east and west axis, and the base, 6 /, will be obtained by subtracting the 
equinoctial shadow, b e, from the measure of amplitude, I e. 



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254 



E. Burgess, etc., 



[iii. 25- 



25. . . . Multiply the sines of co-latitude and of latitude re- 
spectively by the equinoctial shadow and by twelve, 

26. And divide by the sine of declination ; the results are the 
hypothenuse when the sun is on the prime vertical {samaman- 
data). When north declination is less than the latitude, then 
the mid-day hypothenuse {grava), 

27. Multiplied by the equinoctial shadow, and divided by the 
mid-day measure of amplitude (agrd), is the hypothenuse. . . . 

Here we have two separate and independent methods of finding the 
hypothenuse of the east and west shadow cast by the sun at the moment 
when he is upon the prime vertical. In connection with the second of 
the two are stated the circumstances under which alone a transit of the 
sun across the prime vertical will take place : if his declination is south, 
or if, being north, it is greater than the latitude, his diurnal revolution 
will be wholly to the south, or wholly to the north, of that circle. 

The first method is illustrated by the following figures. Let V C" 
(Fig. 12) be an arc of the prime ver- 
tical, V being the point at which the 
sun crosses it in his daily revolution ; 
and let C be the centre ; then V C 
is radius, and VC the sine of the 
sun's altitude; and, C b being the 
gnomon, b v will be the shadow, and 
C'tr its hypothenuse. But, by simi- 
larity of triangles, 

VC:VC'::C'6:C'v 

Again, in the other figure (Fig. 13) — of which the general relations 
arc those of Fig. 8 
(p. 232)— AD being 
the projection of the 
circle of the sun's 
diurnal revolution, 
and the point at 
which it crosses the 
prime vertical being 
seen projected in V, 
V C is the sine of the 
sun's altitude at that 
point. But VCB 
and ECU are sinii- • 
lar triangles, the an- 
gles BVC and CEII 
being each equal to 
the latitude ; nencc 
VC:EC::BC:CH 

Now the first of 
these ratios is — since 
EC equals VC, both 
being radius — the 
same with the first 




Fig. 18. 




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iii. 31.] S&rya-Siddhdnta. 255 

in the former proportion ; and therefore 

BC:CH::C'o:C'v 
or sin decl. : sin lat. : : gnom. : hyp. pr. vert. shad, 

but sin lat. : cos. lat. : : eq. shad. : gnom. 

therefore, by combining terms, 

sin. decl. : cos. lat. : : eq. shad. : hyp. pr. vert shad. 
and the reduction of the first and third of these proportions to the 
form of equations gives the rules of the text 

The other method of finding the same quantity is an application of 
the principle demonstrated above, under verse 7, that, with a given dec- 
lination, the measure of amplitude of any shadow is to that of any other 
shadow as the hypothenuse of the former to that of the latter. Now 
when the sun is upon the prime vertical, the shadow falls directly 
eastward or directly westward, and hence its extremity lies in the east 
and west axis of the dial, and its measure of amplitude is equal to 
the equinoctial shadow. The noon measure of amplitude is, accord- 
ingly, to the hypothenuse of the noon shadow as the equinoctial shadow 
to the hypothenuse of the shadow cast when the sun is upon the prime 
vertical. 

27. ... If the sine of declination of a given time be multiplied 
by radius and divided by the sine of co-latitude, the result is the 
sine of amplitude ((^arndurrrikd). 

28. And this, being farther multiplied by the hypothenuse of 
a given shadow at that time, and divided by radius, gives the 
manure of amplitude (agrd), in digits (angula), etc. . . . 

The sine of the sun's amplitude is found — his declination and the 
latitude being known — by a comparison of the similar triangles ABC 
and C EH (Fig. 13), in which 

HE:EC::BC:CA 
or cos. lat : R : : sin. decl. : sin. ampl. 

And the proportion upon which is founded the rule in verse 28 — name- 
ly, that radius is to the sine of amplitude as the hypothenuse of a given 
shadow to the corresponding measure of amplitude— has been demon* 
strated under verse 7, above. 

28 If from half the square of radius the square of the 

sine of amplitude (agrajyS) be subtracted, and the remainder 
multiplied by twelve, 

20. And again multiplied by twelve, and then farther divided 
by the square of the equinoctial shadow increased by half the 
square of the gnomon — the result obtained by the wise 

80. Is called the "surd" (karani) : this let the wise man set 
down in two places. Then multiply the equinoctial shadow by 
twelve, and again by the sine of amplitude, 

81. And divide as before: the result is styled the "fruit" 
(jphala). Add its square to the " surd," and take the square root 
of their sum ; this, diminished and increased by the " fruit," for 
tbeaofltthern and northern hemispheres, 



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258 



Et Bwrg&pj etc.. 



[iii.S2- 



82. Is the «ne of altitude (pmbu) of the aAuthem intermediate 
directions (fcwftj) ; arid equally; 'whether* the sun's revolution 
take place to the soiith or to the fcorth of the gnomon (canku}-^ 
only, in the latter case, the sine of altitude is that of the north- 
ern intermediate directions. 

38. The square root of the difference of the squares of that 
and of radius is styled the sine of zenith-distance (arg.) If, then, 
the sine of zenith-distance and radius, be multiplied respectively 
by twelve, and divided by the sine of altitude, 

34. The results are thesnado^r and hypothenuse at the angles 
(kona), under the given circumstances of time and place. . . . 

The method taught in this passage of finding, with a given declina- 
tion and latitude, the sine of the sun's altitude at the moment when he 
crosses the south-east and south-west vertical circles, or when the shadow 
of the gnomon is thrown toward the angles (Jcona) of the circumscribing 
square of the dial, is, when stated algebraically, as follows : 

^gn.*+eq. sh.* 
eq. sh. X gn. X sin ampl. . .. 

ign.*-f-eq. sh.* 
\Zaurd -\- fruit* + fruit = sin alt., declination being north, 
vsurd + fruit* — fruit = sin alt., declination being south. 

It is at once apparent that a problem is here presented more compli- 
cated and difficult of solution than any with which we have heretAre 
had to do in the treatise. The commentary gives a demonstration of 
it, in which, for the firet time, the .notation and processes of the 
Hindu algebra are introduced, and with these we are not sufficiently 
familiar to be able to follow, the course of the demonstration. The 
problem, however, admits of solution without the aid of mathematical 
knowledge of a higher character than has been displayed in the pro- 
cesses .already explained; by means, namely, of the consideration of 
right-angled triangles, situated in the same plane, and capable of being 
represented by a single figure. We give below such a solution, which, 
we are persuaded, agrees in ail its 
main features with the process by 
which the formulas of the text were 
originally deduced. 

Let ZEK be the south-eastern 
circle of altitude, from the zenith, Z, 
to the horizon, K ; let £ be its in- 
tersection with the equator, and t) 
the position of the sun ; and let 
<3 b represent the gnomon. 

Since e is in the line of the equi- 
noctial shadow (see above; v. 7), 
and since be makes an angle of 45° 
with either axis of the dial, we 
have &e f =2 eq; sh.», and C« a 32 
C 6» +b e* = gn.» +2 eq. sh.* 




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3i.34.] SAtya^SKchMnia. 257 

In like manner, de 2 =2 meat, ampl, 2 But the similar triangles 
Cde and CDS' give Crf a :rf« 2 :;CD 2 :DE' 2 ; which, by halving 
the two consequents, and observing the constant relation of C d to the 
measure of amplitude (see above, under v. 7), gives R 2 : sin ampl. 2 : : 
R 2 : JD E' 2 : whence *D E' 2 = sin ampl. 2 , or D E' 2 = 2 sin ampl 2 

Now the required sine of altitude is I) G, and D G=D H-j-H G= 
DH+IJ. And, obviously, the triangles DHI, DIE', EFC, IJC, 
and C b e are all similar. TTien, from Dfl I and C b ^ we derive 

DH:DI::6e:Ce 
from DIE' and Cbe, DI : DE' ::Cb:Ce 
and, by combining terms, D H : D W : : b eXC b;Ce* 
t. ^tt y«.eq.ih.XjCn.xy2.Mnampl. eq.atLXgn.Xsio ampl _ . A 

Whence DH= 3 r-p- r-r £-53 -i-r ~- r~-= *""*• 

gn*+2eq.ih.* ign.*-f eq. th.* 

Again, from DHI and EFC, we derive 

IH*:DI f ::EF»:EC* 
from IJC and EFC, IJ f : IC* ::EF» :EC« 
whence, by adding the terms of the equal ratios, and observing that 
IH'+IJ*=JH», and DI«+IC*=DC»=EC*, we have 

JH»:EC»::EF*:EC» 
or JH*=EF* . Hence IJ»=tf H«-IH»==EF f -IH*=EF«-DI*+DH f 
But from EFCand Cbe are derived 

C« 2 :Co 2 ::EC 2 :EF 2 
from DIE' and Cbe, Ce 2 :C6 2 : :DE' 2 :DI 2 

whence EF>= ,and J)I«= — ^ — , end EF»-DP= V (fiT*— 

that is to say, 

gn.H-2 eq. th.* igo.*+ «q. A.* 

But* as was shown a bove, I J«=E F 2 -D I 2 +I> H 2 =surd+fru5t 2 
and VWd+&uit 2 + fruit= I J+ D H=D G = sine of altitude. 

When declination is south, so that the sun crosses the circle of alti- 
tude at D*, I H', the equivalent of D H, is to be subtracted from IJ, to . 
give D' G / , the sine of altitude. 

The correctness of the Hindu formulas may likewise be briefly and 
succinctly demonstrated by means of our modern methods. Thus, let 
PZS (Fig. 15) be a spherical triangle, 
of which the three angular points are 
P, the pole, Z, the zenith, and S, the 
place of the sun when upon the south- 
east or the south-west vertical circles; 
PZ, then, is the co-latitude, ZS the 
zenith-distance, or co-altitude, and P S 
the co-declination ; and the angle PZS is 135° ; the problem is, to find 
the sine of the complement of ZS, or of the sun's altitude. By spheri- 
cal trigonometry, cos S P=cos Z S cos Z P+sin Z S sin Z P cos Z. Di- 
Tiding by sin ZP, and observing that cos SP-J-sin ZP=sin decL-J-coa 
lat= sine of amplitude, we have sin ampler sin alt. tan lat+cos alt cos 
135°. If, now, we represent sin ampl. by a, tan lat by 6, cos 185° by 
wot- vi, 33 




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258 IB. Burgess, etc., [iii. 34- 

— \/±, sin alt. by *, and cos alt by Vl — «*, we have a 8 - 2a b x+b* **= 
J(l— * 2 ); and, by reduction, a; 2 — ttTs g= -14,52 * Representing, 

again, j^-^ by /, and|^5 by *, and reducing, we have *=/+ 

V/^+i. But/ is evidently the same with the "fruit,'' since b, or tan 
, ,,, ,. aft eq.sh.xgn.Xsin.ampl 

lat, equals eq. sh.-J-gnom- and therefore fl . . =-? ^-r-i r-x- • 

^ ^ * ° ^ i+* 8 ^gnom. 8 -f-eq. sh. 8 

, . , .v ..,., „ ,„, +-«* (*R 8 -sin*ampI.)Xgn* 

and s is also the same with the "surd," for ttt;=^ rr*i4--' 

i+b |gnom.*+eq. sh. s 

If, the latitude being north, we consider the north direction as posi- 
tive, b will be positive. The value of/, given above, will then evidently 
be positive or negative as the sign of a is plus or minus. But a, the 
sine of amplitude, is positive when declination is north, and negative 
when declination is south. Hence / is to be added to or subtracted 
from the radical, according as the sun is north or south of the equator, 
as prescribed by the Hindu rule. A minus sign before the radical would 
correspond to a second passage of the sun across the south-east and 
north-west vertical circle ; which, except in a high latitude, would take 
place always below the horizon. 

The construction of the last part of verse 32 is by no means clear, yet 
we cannot question that the meaning intended to be conveyed by it is 
truly represented by our translation. When declination is greater than 
north latitude, the sun's revolution is made wholly to the north of the 
prime vertical, and the vertical circles which he crosses are the north- 
east and the north-west. The process prescribed in the text, however, 
gives the correct value for the sine of altitude in this case also. For, 
in the triangle S ZP (Fig. 15), all the parts remain the same, excepting 
that the angle PZS becomes 45°, instead of 135°: but the cosine of 
the former is the same as that of the latter arc, with a difference only 
of sign, which disappears in the process, the cosine being squared. 

The sine of altitude being found, that of its complement* or of zenith- 
distance, is readily derived from it by the method of squares (as above, 
in w. 16, 17). To ascertain, farther, the length of the corresponding 
shadow and of its hypothenuse, we make the proportions 

sin alt. : sin zen. dist. : : gnora. : shad, 
and sin alt, : R : : gnom. : hyp. shad. 

In this passage, as in those that follow, the sine of altitude is called 
by the same name, fanku, u staff," which is elsewhere given to the 
gnomon : the gnomon, in fact, representing in all cases, if the hypothe- 
nuse be made radius, the sine of the suits altitude. The word is fre- 
quently used in tab sense in the modern astronomical language : thus 
V C (Fig. 13, p. 254), the sine of the sun's altitude when upon the 
prime vertical, is called the mmamandalafanku, " prime vertical staff," 
and B T, the sine of altitude when the sun crosses the unmandala, or 
east and west hour-circle, is styled the unmandalafanku : of the latter 
line, however, the Surya-Siddhanta makes no account We are sur- 
prised, however, not to find a distinct name for the altitude, as for its 
complement, the zenith-distance : the sine of the latter might with very 



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iii. 35.] Sdrya-Sid^dnta. 259 

nearly the same propriety be called the " shadow," as that of the former 
the "gnomon." The particular sine of altitude which is the result of 
the present process is commonly known as the konapanku, from the 
word kona, which, signifying originally u angle," is used, in connection 
with the dial, to indicate the angles of the circumscribing square (see 
Fig. 9, p. 241), and then the directions in which those angles lie from 
the gnomon. The word itself is doubtless borrowed from the Greek 
yaw to, the form riven to it being that in which it appears in the com* 
pounds TQiyvvoy (Sanskrit Irikona), etc. Lest it seem strange that the 
Hindus should have derived from abroad the name for so familiar and 
elementary a quantity as an angle, we would direct attention to the 
striking fact that in that stage of their mathematical science, at least, 
which is represented by the Surya-Siddhanta, they appear to have made 
no use whatever in their calculations of the angle : for, excepting in 
this passage (v. 34) and in the term for " square " employed in a pre- 
vious verse (v. 5) of this chapter, no word meaning "angle" is to be 
met with anywhere in the text of this treatise. The term drf, used to 
signify "zenith-distance" — excepting when this is measured upon the 
meridian ; see above, under w. 14-16 — means literally "sight :" in this 
sense, it occurs here for the first time : we have had it more than once 
above with the signification of " observed place," as distinguished from 
a position obtained by calculation. In verse 32, fan&u might be under- 
stood as used in the sense of " zenith," yet it has there, in truth, its own 
proper signification of " gnomon ;" the meaning being, that the sun, in 
the cases supposed, makes his revolution to the south or to the north of 
the gnomon itself, or in such a manner as to cast the shadow of the 
latter, at noon, northward or southward. * One of the factors in the cal- 
culation is styled karant, "surd" (see Colebrooke's Hind. Alg., p. 145), 
rather, apparently, as being a quantity of which the root is not required to 
be taken, than one of which an integral root is always impossible ; or, it 
may be, as being the square of a line which is not, and cannot be, drawn. 
The term translated "fruit" (jphala) is one of very frequent occurrence 
elsewhere, as denoting " quotient, result, corrective equation," etc. 

The form of statement and of injunction employed in verses 29 and 
30, in the phrases "the result obtained by the wise," and "let the wise 
man set down," etc., is so little in accordance with the style of our 
treatise elsewhere, while it is also frequent and familiar in other works 
of a kindred character, that it furnishes ground for suspicion that this 
passage, relating to the Jtcmapanku, is a later interpolation into the body 
of the text ; and the suspicion is strengthened by the fact that the pro- 
cess prescribed here is so much more complicated than those elsewhere 
presented in this chapter. 

34. ... If radius be increased by the sine of ascensional differ- 
ence (cam) when declination is. north, or diminished by the same, 
when declination is south, 

35. The result is the day-measure (antyd); this, diminished 
by the versed sine (utkramajyti) of the hour-angle (nata), then 
multiplied bv the day-radius and divided by ramus, is the "di- 
visor" (chedd) ; the latter, again, being Multiplied by the sine of 
co-latitude (Icmiba), and divided 



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260 E. Burgm, efc>, pii$6- 

36. By radio*, gives the sine of altitude ($antofy? subtract its 
sine from that of radius, and the square root of the remainder is 
the sine of zenith*distanoe (efr^): the shadow and itshypothg- 
nuse are found as in the preceding process. 

The object of this process is, to find the sine of the sun's altitude at 
may given hour of the day, when his distance from the meridian* his 
declination, and the latitude, are known. The sun's angular distance 
from the meridian, or the hour-angle, is found, as explained by the com- 
mentary, by subtracting the time elapsed since sunrise, or which is to 
elapse before sunset, from the half day, as calculated by a rule previously 
given (ii. 61*63). From the declination and the latitude the sine of 
ascensional difference (carajyd) is supposed to have been already derived, 
by the method taught in the same passage; as also, from the declina- 
tion (by ii. 60), the radius of the diurnal circle. The successive steps 
of the process of calculation will be made clear by a reference to the 
annexed figure (Fig. 16), taken in connection with Fig. 13 (p. 254), with 
which it corresponds in dimensions and lettering. Let G G' O E repre- 
sent a portion of the plane of the equator, C being its centre, and G £ 
its intersection with the plane of the me- j^ j- 

ridian ; and let A A' B' D represent a cor- 
responding portion of the plane of the 
diurnal circle, as seen projected upon the 
other, its centre and its line of intersection 
with the meridian coinciding with those 
of the latter. Let C G eoual the sine of 
ascensional difference, and A B its corre- 
spondent in the lesser circle, or the earth- 
sine (kigyd or kshityyd ; see above, ii. 61). 
Now let O' be the place of the sun at a 
given time; the angle O'CD, measured 
by the arc of the equator Q 7 E, is the 
hour-angle : from Qf draw Q'Q perpendic- 
ular to C E ; then Q 7 Q is the sine, and 
QE is the versed sine, of Q'E. Add to 
radius, E C, the sine of ascensional difference, C G ; their sum, E G — 
which is the equivalent, in terms of a great circle, of D A, that part of 
the diameter of the circle of diurnal revolution which is above the 
horizon, and which consequently measures the length of the day — is 
the day-measure (anty&y From EG deduct E Q, the versed sine of the 
hour-angle ; the remainder, G Q, is the same quantity in terms of a great 
circle which A O is in terms of the diurnal circle : hence the reduction 
of G Q to the dimensions of the lesser circle, by the proportion 

CE:BD::GQ:AO 
gives us the value of A ; to this the text gives the technical name of 
" divisor » (cheda). But, by Fig. 13, 

CE:EH::AO:OR 
hence R, which is the sine of the sun's altitude at the given time, 
equals A O, the " divisor," multiplied by EH, the cosine of latitude, and 
divided by C E, ot radius. * 




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lit 41.] Stirya-Sxidhfoita. 281 

The processes &r derivtog from* the sine of altitude that of zenith- 
distance, and from both the length of the corresponding shadow and its 
hypothenuae, are precisely the same as in the last problem. 

For the meaning of antyb — which, for lack of a better term, we have 
translated "day-measure" — see above, under verse 7. The word natc^ 
by which the hour-angle is designated, is the same with that employed 
above with the signification of u meridian zenith-distance f see the note 
to verses 14-17. *■ 

37. If radius be multiplied by a given shadow, and divided 
by the corresponding hypotheiuae, the result is the sine of 
senith-distance (dfc): the square root of ti& difference between 
the square of that and the square of radius 

88. Is the sine of altitude (yanku); which, multiplied by 
radius and divided by the trine of cxwatitttde (lamba), gives the 
"divisor" (cheda) : multiply the lattei* by radius, and divide by 
the radius of the diurnal circle, * 

89. And the quotient is the sine of the sun's distance from the 
horizon (unnata)] this, then, being subtracted from the day- 
measure (antyd), and the remainder turned into arc by means of 

ee table of versed sines, the final result is the hour-angle (naia), 
respirations (asu), east or west 

» ; The process taught in these verses is precisely the converse of the 
jjjme described in the preceding passage. The only point which calls for 
Jkrther remark in connection with it is, that the line GQ (Fig. 16) is in 
forse 39 called the "sine of the unnata* By this latter term is desig- 
nated the opposite of the hour-angle (nata) — that is to say, the sun s 
lingular distance from the horizon upon his own circle, O A', reduced to 
pne, or to the measure of a great circle. Thus, when the sun is at O, 
lis hour-angle (nata\ or the time till noon, is Q*E; his distance from 
the horizon (unnata), or the time since sunrise, is Q* G'. But G Q is 
.with no propriety styled the sine of G'Q'; it is not itself a sine at all, 
■and the actual sine of the arc in question would have a very different 
value. 

40. Multiply the sine of co-latitude by any given measure of 
amplitude (agrd), and divide by the correeponding hvpothenuse 
in digits; the result is the sine of declination. This, again, is 
to be multiplied by radius, and divided by the sine of greatest 
declination ; 

41. The quotient, converted into arc, is, in signs, etc., the sun's 
place in the quadrant; by means of the quadrants is then found 
the actual longitude of the sun at that point. . . . 

By the method taught in this passage, the sun's declination, and, 
through that, his true and mean longitude, may, the latitude of the ob- 
server being known, be found from a single observation upon the shadow 
at any hour in the day. The declination is obtained from the measure 
of amplitude and the hypothenuse of the shadow, in the following 



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262 M Burgess, &-, [Hi. 41- 

manner : first, as was shown in connection with Terse 1 of this chapter, 
hyp. shad. : meas. ampl. : : £ : C A (Fig. 13, p. 264) 
but EC:CA::EH:BC 

therefore hyp. shad. : meas. ampl. : : £ H : B C 

B C, the sine of declination, being thus ascertained, the longitude is de- 
duced from it as in a previous process (see above, vv. 17-20). 

41. . . . Upon 3 given day, the distances of three baaes, at 
noon, in the forenoon, and in the afternoon, being laid off, 

42. From the point of intersection of the lines drawn between 
them by means of two fish-figures, (maUyd), and with a radius 
touching the three points, is describea the path of the shadow. . . 

This method of drawing upon the face of the dial the path which 
will be described by the extremity of the shadow upon a given day pro- 
ceeds upon the assumption that that path will be an arc of a circle — an 
erroneous assumption, since, excepting within the polar circles, the path 
of the shadow is always a hyperbola, when the sun is not in the equator. 
In low latitudes, however, the difference between the arc of the hyper- 
bola, at any point not too far from the gnomon, and the arc of a circle, 
is so small, that it is not very surprising that the Hindus should have 
overlooked it. The path being regarded as a true circle, of course it 
can be drawn if any three points in it can be found by calculation : and 
this is not difficult, since the rules above given furnish means of ascer- 
taining, if the sun's declination and the observer's latitude be known, 
the length of the shadow and the length of its base, or the distance of 
its extremity from the east and west axis of the dial, at different times 
during the day. One part of the process, however, has not been provi- 
ded for in the rules hitherto given. Thus (Fig. 9, p. 241), supposing 
(J, m, and I to be three points in the same daily path of the shadow, we 
require, in order to lay down I and m, to know not only the bases I fc", 
wi&'", but also the distances 56", bb m . But these are readily found 
when the shadow and the base corresponding to each are known, ox 
they may be calculated from the sines of the respective hour-angles. 

The three points being determined, the mode of describing a circle 
through them is virtually the same with that which we should employ : 
lines are drawn from the noon-point to each of the others, whicn are 
then, by fish-figures (see above, under w. 1-6), bisected by other lines at 
right angles to them, and the intersection of the latter is the centre of 
the required circle. 

42. . . . Multiply by the day-radius of three signs, and divide 
by their own respective day-radii, 

48. In succession, the sines of one, of two, and of three signs; 
the quotients, converted into arc, being subtracted, each from the* 
one following, give, beginning with Aries, the times of rising 
(udaydsavas) at LankS; 

44. Namely sixteen hundred and seventy, seventeen hundred 
and ninety-five, and nineteen hundred and thirty-five respira- 
tions. . And these t diminished each by its portion of ascensional 



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iii.45.] S&rya-Siddhdnta. 

difference (cardkhanda), as calculated for a given place, are the 
times of rising at that place. 

45. Invert them, and add their own portions of ascensional 
difference inverted, and the sums are the three signs beginning 
with Cancer : and these same six, in inverse order, are the other 
six, commencing with Libra. 

The problem here is to determine the "times of rising" (uday&savas) 
of the different signs of the ecliptic — that is to say, the part of tie 5400 
respirations (asavas) constituting a quarter of the sidereal day, which 
each of the three signs making up a quadrant of the ecliptic will occupy 
in rising {udaya) above the horizon. And in the first place, the times 
of rising at the equator, or in the right sphere — which are the equiva- 
lents of the signs m right ascension — are found as follows : 

Let ZN (Fig. 17) be a quadrant of the solstitial colure, A N the pro- 
jection upon its plane of the equinoctial colure, A Z of the equator, and 
AC of the ecliptic; and let A, T, G, and 
C be the projections upon A C of the initial ^ t ^ m ^^^ 
points of the first four signs ; then A T is 
the sine of one sign, or of 30°, AG of two 
signs, or of 60°, and A C, which is radius, 
the sine of three signs, or of 90°. From 
T, G, and C, draw T t, Qg, C c, perpendicu- 
lartoAN. Then A T t and A C c are simi- 
lar triangles, and, since A C equals radius, 
R:Cc::AT:T* 

But the arc of which T t is sine, is the 
same part of the circle of diurnal revolu- 
tion of which the radius is t P, as the re- 
quired ascensional equivalent of one sign is 
of the equator : hence the sine of the latter, which we may call a, is 
found by reducing T t to the measure of a great circle, which is done by 
the proportion 

f?:R::Tt:sin* 
Combining this with the preceding proportion, we have, 
**':Cc::AT:sins 

Again, to find the ascensional equivalent of two signs, which we will 
call y, we have first, by comparison of the triangles A6^ and A Cc, 

R:Cc::AG:G^ 
and gg* :R::Gg : sin y 

therefore, as before, ^^:Cc::AG:siny 

Hence, the sines of the ascensional equivalents of one and of two signs 
respectively are equal to the sines of one and of two signs, A T and 
AG, multiplied by the day-radius of three signs, Cc, and divided each 
bv its own day-radios, tP and gg*; and the conversion of the sines thus 
obtained into arc gives the ascensional equivalents themselves. The 
rule of the text includes also the equivalent of three signs, but this is so 
obviously equal to a quadrant that it is unnecessary to draw out the 
process, all the terms in the proportions disappearing except radius. 




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264 R Burgess, etc., [iii.45- 

Upon working out the process, by means of the table of sines given 
in the second chapter (w. 15-22), and assuming the inclination of the 
plane of the ecliptic to be 24° (ii. 28), we find, by the rule riven above 
(ii. 60), that the day-radii of one, of two, and of three ■Jbiiifor tt\ gg*, 
Cc, are 3366', 3216', and 3140' respectively, and that the sines of * and 
y are 1604' and 2907', to which the corresponding arcs are 27° 50' and 
57° 45', or 1670' and 3465'. The former is the ascensional equivalent 
of the first sign ; subtracting it from the latter gives that of the second 
sign, which is 1795', and subtracting 3465' from a quadrant, 5400', 
gives the equivalent of the third sign, which is 1935' — all as stated in 
the text 

These, then, are the periods of sidereal time which the first three 
signs of the ecliptic will occupy in rising above the horizon at the equa- 
tor, or in passing the meridian of any latitude. It is obvious that the 
same auantities, in inverse order, will be the equivalents in right ascen- 
sion of the three following signs also, and that the series of six equiva- 
lents thus found will belong also to the six signs of the other half of the 
ecliptic. In order, now, to ascertain the equivalents of the signs in 
oblique ascension, or the periods of sidereal time which they will occupy 
in rising above the horizon of a given latitude, it is necessary first to 
calculate, for that latitude, the ascensional difference (cara) of the three 
points T, G, and C (Fig. 17), which is done by the rule given in the last 
chapter fvv. 61, 62). We have calculated these quantities, in the Hindu 
method, tor the latitude of Washington, 38° 54', and find the ascensional 
difference of T to be 578', that of G 1061', and that of C 1263'. The 
manner in which these are combined with the equivalents in right 
ascension to produce the equivalents in oblique ascension may bo ex- 
plained by the following figure (Fig. 18), which, although not a true 
projection, is sufficient for the purpose p . ._ 

of illustration. Let A C S be a semi- ^* 

circle of the ecliptic, divided into its 
successive signs, and A S a semicircle 
of the equator, upon which AT', T' G', 
etc., are the equivalent* of those signs 
in right ascension ; and let t, g y etc., be 
the points which rise simultaneously 
with T, G, etc. Then t T and v V, the 
ascensional difference of T and V, are 
578', g G' and W are 1061', and cC 
is 1263'. Then A *, the equivalent in 
oblique ascension of AT, equals A T' - 
t T', or 1092'. To find, again, the value 
of tg, the second equivalent, the text 
directs to subtract from T' G' the differ- 
ence between tV and ^G', which is 
called the carokhanda, "portion of ascensional difference" — that is to 
say, the increment or decrement of ascensional difference at the point 
G as compared with T. Thus 

^=T'G'-(^a'-*TO=T'G / +«T / - 5 ra / =s<G'-^G / =lSll' 
and ^e=G'0 / -(cO / - 5 rG') = G / 0'-r^»'-eC / =5rO / -cC / =17S8' 




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iiit 49.] 



S&rya»Siddhdnta. 



266 



Farther, to find the oblique equivalents in the second quadrant, we 
are directed to invert the right equivalents, and to add to each its own 
carakhanda, decrement of ascensional difference. Thus 
c/==C'L'+(<?O'-/L0i=cL'-IL / =21sr 
/ v = L' V'+ (/ L'-v V')= J V'- tr V'*r 2278' 
and finally, t>8~V'S + *V'=2248'. 

It is obvious without particular explanation that the arcs of oblique 
ascension thus found as the equivalents, in a given latitude, of the first 
six signs of the ecliptic, are likewise, in inverse order, the equivalents of 
the other six. We have, then, the following table of times of rising 
[uday&savai), for the equator and for the latitude of "Washington, of au 
the divisions of the ecliptic : 



Equivalent* in Right and Oblique Ascension of the Sign* of the Ecliptic. 


bt*. 


Sign. 
Nam*. 


Equivalent 

in 

Eight AffMAiioa. 


Lat.of 

Aicem. 

Dift 


Washington. 

Sqolv. In 
ObL Atcfofioo. 


Sign. 


No. 


i. 

3. 
4. 
5. 
6. 


Aries, matha, 
Taurus, vftAan, 
Gemini, mitkuna, 
Cancer, karkata, 
Leo, tinha, 
Virgo, kanyd, 


'or p. 
1670 
1795 
1935 
1935 
1795 
1670 


'or P. 

578 
1061 
ia63 
1 061 

578 


'or p. 
1092 
l3ia 
i 7 33 
ai37 
2278 
2248 


Pisces, mint, 
Aquarius, fatmbha, 
Gapricornu8,niaA;ara, 
Sagittarius, dhanus, 
Scorpio, dli, 
Libra, tutd t 


12. 
II. 

10. 
9- 
8. 

7- 



For the expression " at Lanka," employed inverse 48 to designate 
the equator, see above, under i. 62. 

. 46. From the longitude of the sun at a given time are to be 
calculated the ascensional equivalents of the parts past and to 
come of the sign in which he is: they are equal to the number 
* _ \ traversed and to be traversed, multiplied by the as- 
>hal equivalent (udaydsavas) of the sign, and divided by 

Then, from the given time, reduced to respirations, sub- 

f the equivalent, in respirations, of the part of the sign to 

lie, and also the ascensional equivalents (lagndsavas) of the 

jfowing signs, in succession — so likewise, subtract the equiva- 

of the part past, and of the signs past, in inverse order; 

.If there be a remainder, multiply it by thirty and divide 

jrthe equivalent of the unsubtracted sign: add the quotient, in 

to the whole signs, or subtract it from them : the result 

ie point of the ecliptic (lagna) which is at that time upon the 

>n (kshitija). 

So, from the east or west hour-angle (naia) of the sun, in 
„^_s, having made a similar calculation, by means of the equiv- 
alents in right ascension (fankodaydsams), apply the result as an 
additive or subtraotive equation to the sib s longitude: the re- 
sult is the point of the ecliptic then upon the meridian (madhyar 
lagna). 



VOL. VI. 



34 



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266 JR Burgess, etc., [iii. 49- 

The word lagna means literally "attached to, connected with,' 9 and 
hence, " corresponding, equivalent to." It is, then, most properly, and 
likewise most usually, employed to designate the point or the arc of the 
equator which corresponds to a given point or arc of the ecliptic. In 
such a sense it occurs in this passage, in verse 47, where lagnatavas is 
precisely equivalent to udayasavai, explained in connection with the 
next preceding passage; also below, in verse 50, and in several other 
places. In verses 48 and 49, however, it receives a different significa- 
tion, being taken to indicate the point of the ecliptic which, at a given 
time, is upon the meridian or at the horizon ; the former being called 
lagnam kshiHje, "lagna at the horizon' 9 — or, in one or two cases else* 
where, simply lagna — the other receiving the name of madkyalagna, 
"meridian-tapta." 

The rules by which, the sun's longitude and the hour of the day being 
known, the points of the ecliptic at the horizon and upon the meridian 
are found, are very elliptically and obscurely stated in the text ; our 
translation itself has been necessarily made in part also a paraphrase and 
explication of them. Their farther illustration may be best effected by 
means of an example, with reference to the last figure (Fig. 18). 

At a given place of observation,, as Washington, let the moment of 
local time—reckoned in the usual Hindu manner, from sunrise — be 18° 
12 T 8P, and let the longitude of the sun, as corrected by the precession, 
be, by calculation, 42°, or 1 B 12° : it is required to know the longitude 
of the point of the ecliptic (lagna) then upon the eastern horizon. 

Let P (Fig. 18) be the place of the sun, and H h the line of the hori- 
zon, at the given time ; and let p be the point of the equator which rose 
with the sun ; then the arc p h is equivalent to the time since sunrise, 
18 11 12* 8p, or 6555P. The value of tg y the equivalent in oblique ascen- 
sion of the second sign TG, in which the sun is, is given in the table 
presented at the end of the note upon the preceding passage as 1812'. 
To find the value of the part of itpg we make the proportion 

TQ:TQ::tg:pg 
or 30°:18°::1312P:Wp 

From ph, or 6555P, we now subtract pg y V87p, and then, in succession, 
the ascensional equivalents of the following signs, GC and CL — that is, 
gc f or 1733P, and cl, or 213?p — until there is left a remainder, /A, or 
I898P, which is less than the equivalent of the next sign. To this re- 
mainder of oblique ascension the corresponding arc of longitude is then 
found by a proportion the reverse of that formerly made, namely 

Z*:/A::LV:LH 
or 2278P : 1808P : : 30° : 25° 

The result thus obtained being added to A L, or 4 a , the sum, 4 s 25°, or 
145°, is the longitude of H. 

The arc pg is called in the text bhogy&savas, " the equivalent in respi- 
rations of the part of the sign to be traversed," while tp is styled bh*k* 
fdstnas, " the respirations of the part tnaversed." 

I£ on the other haud^l were desired to arrive at the same result by 
reckoning in the opposite direction from the sun to the horizon, either 
on account of the greater proximity of the two in that direction, or for 



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iii. 51.] S&rya-SMdhdnta. 267 

any other reason, the manner of proceeding would be somewhat differ- 
ent Thus, if AH (Fig. 18) were the sun's longitude, andjpP the line 
of the eastern horizon, we should first find Ajp, by subtracting the part 
of the day already elapsed from the calculated length of the day (this 
step is, in the text, omitted to be specified) ; from it we should then 
subtract the bhukt&savas, I A, and then the equivalents of the signs 
through which the sun has already passed, in inverse order, until there 
remained only the part of an equivalent, pg, which would be converted 
into the corresponding arc of longitude, P G, m the same manner as 
before: and the subtraction of PG from AG would give AP, the 
longitude of the point P. 

Bui again, if it be required to determine the point of the ecliptic 
which is at any given time upon the meridian, the general process is the 
same as already explained, excepting that for the time from sunrise is 
substituted the time until or since noon, and also for the equivalents in 
oblique ascension those in right ascension, or, in the language of the 
text) the u times of rising at Lanka" (tonJcoddyctoxwu); since the me- 
ridian, like the equatorial horizon, cuts the equator at right angles. 

It will be observed that all these calculations assume the increments 
longitude of to be proportional to those of ascension throughout each 
sign : in a process of greater pretensions to accuracy, this would lead to 
errors of some consequence. 

The use and value of the methods here taught, and of the quantities 
found as their results, will appear in the sequel (see ch. v. 1-9 ; vii. 7 ; 
ix. 5-11 ; x. 2). 

The term kthitija, by which the horizon is designated, may be under- 
stood, according to the meaning attributed to k$h*H (see above, under 
ii 61-dS), either as the " circle of situation" — that is, the one which is 
dependent upon the situation of the observer, varying with every change 
of place on his part — or as the "earth-circle," the one produced by the 
intervention of the earth below the observer, or drawn by the earth 
upon the sky. Probably the latter is its true interpretation.^ 

50. Add together the ascensional equivalents, in respirations, 
of the part of the sign to be traversed by the point having lees 
longitude, of the part traversed by that having greater longitude, 
and of the intervening signs — thus is made the ascertainment of 
time (kdlasddhana), 

51. When the longitude of the point of the ecliptic upon the 
horizon (lagna) is less than that of the sun, the time is in the 
latter part of the night; when greater, it is in the day-time; 
when greater than the longitude of the sun increased by half a 
revolution, it is after sunset 

Hie process taught in these verses is, in a manner, the convene of 
that which is explained in the preceding passage, its object being to find 
the instant of local time when a given point of the ecliptic will oe upon 
the horizon, the longitude of the sun being also known. Thus (Fig. 18), 
supposing the sun's longitude, A P, to be, at a given time, I s 12° ; it n 
required to know at what time the point H, of which the longitude is 



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268 R Burgess, etc., [iii. 51- 

4 § 25°, will rise. The problem, is, virtually, to ascertain the arc of the 
equator intercepted between p } the point which rose with the sun, and 
A, which will rise with H, since that arc determines the time elapsed 
between sunrise and the rise of H, or the time in the day at which the 
latter will take place. In order to this, we ascertain, by a process simi- 
lar to that illustrated in connection with the last passage, the bkogyA- 
savas, "ascensional equivalent of the part of the sign to be traversed," 
of the point having less longitude — or pg — and the bhukt&mvas, "as- 
censional equivalent of the part traversed," belonging to H, the point 
having greater longitude — or / h — and add the sum of both to that of 
the ascensional equivalents of the intervening whole signs, g e and c 7, 
which the text calls antaralagn&savas, " equivalent respirations of the 
interval ;" the total is, in respirations of time, corresponding to minutes 
of arc, the interval of time required : it will be found to be 6555?, or 
18 n 12* 3P: and this, in the case assumed, is the time in the day at 
which the rise of H takes place : were H, on the other hand, the posi- 
tion of the sun, 18 n 12* 3* would be the time before sunrise of the same 
event, and would require to be subtracted from the calculated length of 
the day to give the instant of local time. 

It is evident that the main use of this process must be to determine 
the hour at which a given planet, or a star of which the longitude is 
known, will pass the horizon, or at which its "day" (see above, ii. 59- 
68) will commence. A like method — substituting only the equiva- 
lents in right for those in oblique ascension — might be employed in 
determining at what instant of local time the complete day, aJwr&tra, 
of any of tne heavenly bodies, reckoned from transit to transit across 
the lower meridian, would commence : and this is perhaps to be re- 
garded as included also in the terms of verse 50 ; even though the 
following verse plainly has reference to the time of rising, and the word 
lagna, as used in it, means only the point upon the horizon. 

The last verse we take to be simply an obvious and convenient rule 
for determining at a glance in which part of the civil day will take 
place the rising of any given point of the ecliptic, or of a planet occu- 
pying that point. If the longitude of a planet be less than that of the 
sun, while at the same time they are not more than three signs apart — 
this and the other corresponding restrictions in point of distance are 
plainly implied in the different specifications of the verse as compared 
with one another, and are accordingly explicitly stated by the commen- 
tator — the hour when that planet comes to assume the position called 
in the text lagna, or to pass the eastern horizon, will evidently be 
between midnight and sunrise, or in the after part (pesha, literally " re- 
mainder") of the night: if, again, it be more than three and less than 
six signs behind the sun, or, which is the same thing, more than six and 
less than nine signs in advance of him, its time of rising will be between 
sunset and midnight : if, once more, it be in advance of the sun by less 
than six signs, it will rise while the sun is above the horizon. 

The next three chapters treat of the eclipses of the sun and moon, the 
fourth, being devoted to lunar eclipses, and the fifth to solar, and the 
sixth containing directions for projecting an eclipse. 



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iv. 1.] S&rya-Siddhdnta. 269 

CHAPTER IV. 



Contents :-~l, dimensions of the sun and moon; 2-9, measurement of their apparent 
dimensions; 4-6, measurement of the earth's shadow; 6, conditions of the occur- 
rence of an eclipse; 7-8, ascertainment of longitude at the time of conjunction or 
of opposition; a, causes of eclipses; 10-11, to determine whether there will be 
an eclipse, and the amount of obscuration; 12-16, to find half the time of dura- 
tion of the eclipse, and half that of total obscuration; 16-17, to ascertain the 
times of contact and of separation, and, in a total eclipse, of immersion and 
emergence; 18-21, to determine the amount of obscuration at a given time; 
22-23, to find the time corresponding to a given amount of obscuration ; 24-26, 
measurement of the deflection of the ecliptic, at the point occupied by the 
eclipsed body, from an east and west line; 26, correction of the scale of projec- 
tion for difference of altitude. 

1. The diameter of the sun's disk is six thousand five hun- 
dred yojanas ; of the moon's, four hundred and eighty. 

We shall see, in connection with the next passage, that the diameters 
of the sun and moon, as thus stated, are subject to a curious modifica- 
tion, dependent upon and representing the greater or less distance of 
those bodies from the earth ; so that, in a certain sense, we have here 
only their mean diameters. These represent, however, in the Hindu 
theory — which affects to reject the supposition of other orbits than such 
as are circular, and described at equal distances about the earth — the 
true absolute dimensions of the sun and moon. 

Of the two, only that for the moon is obtained by a legitimate pro- 
cess, or presents any near approximation to the truth. The diameter of 
the earth being, as stated above (i. 59), 1600 yojanas, that of the moon, 
480 yojanas, is .3 of it : while the true value of the moon's diameter in 
terms of the. earth's is .2716, or only about a tenth less. An estimate 
so nearly correct supposes, of course, an equally correct determination 
of the moon's horizontal parallax, distance from the earth, and mean 
apparent diameter. The Hindu valuation of the parallax may be de- 
duced from the value given just below (v. 3), of a minute on the moon's 
orbit, as 15 yojanas. Since the moon's horizontal parallax is equal to 
the angle subtended at her centre by the earth's radius, and since, at 
the moon's mean distance, 1' of arc equals 15 yojanas, and the earth's 
radius, 800 yojanas, would accordingly subtend an angle of 58' 20" — the 
latter angle, 53' 20", is, according to the system of the Surya-Siddhanta, 
the moon's parallax, when in the horizon and at her mean distance. 
This is considerably less than the actual value of the quantity, as deter* 
ruined by modern science, namely 67" 1'; and it is practically, in the 
calculation of solar eclipses, still farther lessened by 3' 51", the excess 
of the value assigned to the sun's horizontal parallax, as we shall see 
farther on. Of the variation in the parallax, due to the varying distance 
of the moon, the Hindu system makes no account : the variation is actu- 
vol. vi. 35 



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270 M Burgess, etc, [iv. 1- 

ally nearly 8', being from 58' 4B", at the apogee, to 61' 24", at jtbe 
perigee. 

How the amount of the parallax was determined by the Hindus — ifj 
indeed, they had the instruments and the skill in observation requisite 
for making themselves an independent determination of it — we are not 
informed. It is not to be supposed, however, that an actual estimate of 
the mean horizontal parallax as precisely 53' 20" lies at the foundation 
of the other elements which seem to rest upon it; for, in the making 
up of the artificial Hindu system, all these elements have been modified 
and adapted to one another in such a manner as to produce certain 
whole numbers as their results, and so to be of more convenient use. 

Prom this parallax the moon's distance may be deduced by the pro- 
portion 

sin 53' 20" : R : : earth's rail. : moon's dist 
or 53'i : 3438' ::8ooy: 5 1,5707 

The radius of the moon's orbit, then, is 51,570 yojanas, or, in terms of 
the earth's radius, 64.47. The true value of the moon's mean distance 
is 59.06 radii of the earth. 
The farther proportion 

3438' : 54oo' :: 57,5707 : 81,0007 

would give, as the value of a quadrant of the moon's orbit, 81,000 yoja- 
nas, and, as the whole orbit, 324,000 yojanas. This is, in fact, the cir- 
cumference of the orbit assumed by the system, and stated in another 
place (xii. 85). Since, however, the moon's distance is nowhere assumed 
as an element in any of the processes of the system, and is even directed 
(xii. 84) to be found from the circumference of the orbit by the false 
ratio of 1 :^/10, it is probable that it was also made no account of in 
constructing the system, and that the relations of the moon's parallax 
and orbit were fixed by some such proportion as 
53' 20" : 36o° : : 8oo7 : 3a4,ooo7 

The moon's orbit being 324,000 yojanas, the assignment of 480 yoja- 
nas as her diameter implies a determination of her apparent diameter 
at her mean distance as 32' ; since 

36o° : 3a' : : 324,0007 : 4807 

The moon's mean apparent diameter is actually 31 ; 7". 

In order to understand, farther, how the dimensions of the sun's orbit 
and of the sun himself are determined by the Hindus, we have to notice 
that, the moon's orbit being 324,000 yojanas, and her time of sidereal 
revolution 27 d .32167416, the amount of her mean daily motion is 
11,858^.717. The Hindu system now assumes that this is the precise 
amount of the actual mean daily motion, in space, of all the planets, 
and ascertains the dimensions of their several orbits by multiplying it 
by the periodic time of revolution of each (see below, xii. 80-90). The 
length of the sidereal year being 365 d .25875648, the sun's orbit is, a» 
stated elsewhere (xii. 86), 4,331,500 yojanas. From a quadrant of this, 
by the ratio 5400 y : 3438', we derive the sun's distance from the^arth, 
689,430 yojanas, or 861.8 radii of the earth. This is vastly less than 
his true distance, which is about 24,000 radii. His horizontal parallax 



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iv. 3.] S&rya-Siddhdnta. 271 

is, of course, proportionally over-estimated, being made to be nearly 4' 
(more exactly, 3' 59".4), instead of 8".6, its true value, an amount so 
small that it should properly have been neglected as inappreciable. 

It is an important property of the parallaxes of the sun and moon, 
resulting from the manner in which the relative distances of the latter 
from the earth are determined, that they are to one another as the mean 
daily motions of the planets respectively : that is to say, 
53' 20" : 3' 59" : : 790' 35" : 59' 8" 

Each is likewise very nearly one fifteenth of the whole mean daily 
motion, or equivalent to the amount of arc traversed by each planet in 
4nadts; the difference being, for the inoou, about 38", for the sun, 
about 3". We shall see that, in the calculations of the next chapter, 
these differences are neglected, and the parallax taken as equal, in each 
case, to the mean motion during 4 nadis. 

The circumference of the sun's orbit being 4,331,500 yojanas, the 
assignment of 6500 yojanas as his diameter implies that his mean appar- 
ent diameter was considered to be 32' 24".8 ; for 

36o Q : 3a' 24".8 : : 4,33i,5ooy : 65ooJ 

The true value of the sun's apparent diameter at his mean distance is 
32' 3".6. 

The results arrived at by the Greek astronomers relative to the paral- 
lax, distance, and magnitude of the sun and moon are not greatly dis- 
cordant with those here presented. Hipparchus found the moon's hori- 
zontal parallax to be 57' : Aristarchus had previously, by observation 
upon the angular distance of the sun and moon when the latter is half- 
illuminated, made their relative distances to be as 19 to 1 ; this gave 
Hipparchus 3' as the sun's parallax. Ptolemy makes the mean dis- 
tances of the sun and moon from the earth equal to 1210 and 59 radii 
of the earth, and their parallaxes 2' 51" and 58' 14" respectively: he 
also states the diameter of the moon, earth, and sun to be as 1,3$, 18$, 
while the Hindus make them as 1,3£, and 13j£, and their true values, 
as determined by modern science, are as 1,3 J, and 41 2 J, nearly. 

2. These diameters, each multiplied by the true motion, and 
divided by the mean motion, of its own planet, give the cor- 
rected (sphuta) diameters. If that of the sun be multiplied by 
the numoer of the sun's revolutions in an Age, and divided by 
that of the moon's, 

3. Or if it be multiplied by the moon's orbit (kakshd), and 
divided by the sun's orbit, the result will be its diameter upon 
the moon's orbit : all these, divided by fifteen, give the measures 
of the diameters in minutes. 

The absolute values of the diameters of the sun and moon being 
stated in yojanas, it is required to find their apparent values, in minutes 
of arc. In order to this, they are projected upon the moon's orbit, or 
upon a circle described about the earth at the moon's mean distance, of 
which circle — since 824,000-7-21,600=15 — one minute is equivalent 
to fifteen yojanas. 



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272 E. Burgess, etc.. [iv. 3- 

The method of the process will be made clear by the annexed fignre 
(Fig. 19). Let E be the earth's place, EMorEm the mean distance of 

Kg. 19. 




the moon, and E S the mean distance of the sun. Let T U equal the 
sun's diameter, 65007. But now let the sun be at the greater distance 
E S' : the part of his mean orbit which his disk will cover will no longer 
be TU, but a less quantity, tu, and tu will be to TU, or T'U', as ES 
to E S'. But the text is not willing to acknowledge here, any more 
than in the second chapter, an actual inequality in the distance of the 
son from the earth at different times, even though that inequality be 
most unequivocally implied in the processes it prescribes : so, instead of 
calculating E S' as well as E S, which the method of epicycles affords 
full facilities for doing, it substitutes, for the ratio of E 8 to ES', the 
inverse ratio of the daily motion at the mean distance E S to that at the 
. true distance E S\ The ratios, however, are not precisely equal. The 
arc am (Fig. 4, p. 211) of the eccentric circle is supposed to be trav- 
ersed by the sun or moon with a uniform velocity. If, then, the motion 
at any given point, as m, were perpendicular to E m, the apparent mo- 
tion would be inversely as the distance. But the motion at m is per- 
pendicular to em instead of Em. The resulting error, it is true, and 
especially in the case of the sun, is not very great It may be added 
that the eccentric circle which best represents the apparent motions of 
the sun and moon in their elliptic orbits, gives much more imperfectly 
the distances and apparent diameters of those bodies. The value of t ti , 
however, being thus at least approximately determined, V u\ the arc of 
the moon's mean orbit subtended by it, is then found by the proportion 
ES : Em (or EM) : : tu : i'u 1 — excepting that here, again, for the ratio 
of the distances, E S and E M, is substituted either that of the whole 
circumferences of which they are respectively the radii, or the inverse 
ratio of the number of revolutions iu a given time of the two planets, 
which, as shown in the note to the preceding passage, is the same 
thing. Having thus ascertained the value of I' u' in yojanas, division 
by 15 gives ns the number of minutes in the arc t'u' y or in the angle 
t'Eu\ 

In like manner, if the moon be at less than her mean distance from 
the earth, as E M', she will subtend an arc of her mean orbit n o, greater 
than N 0, her true diameter ; the value of n o, in yojanas and in minutes, 
is found by a method precisely similar to that already described. 

There is hardly in the whole treatise a more curious instance than 
this of the mingling together of true theory and false assumption in the 
same process, and of the concealment of the real character of a process 
by substituting other and equivalent data for its true elements. 



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iv. $.] S&rya-Sidd/idnta. 278 

We meet for the first time, in this passage, the term employed in the 
treatise to designate a planetary orbit, namely kakthb, literally u border, 
girdle, periphery.' 7 The value finally obtained for the apparent diame- 
ter of the sun or moon, as later of the shadow, is styled its m&na, 



In order to furnish a practical illustration of the processes taught in 
this chapter, we have calculated in full, by the methods and elements of 
the Surya-Siddhanta, the lunar eclipse of Feb. 6th, 1860. Rather, how- 
ever, than present the calculation piecemeal, and with its different pro- 
cesses severed from their natural connection, and arranged under the 
passages to which they severally belong, we have preferred to give it 
entire in the Appendix, whither the reader is referred for it. 

4. Multiply the earth's diameter by the true daily motion of 
the moon, and divide by her mean motion : the result is the 
earth's corrected diameter (s{cct). The difference between the 
earth's diameter and the corrected diameter of the sun 

5, Is to be multiplied by the moon's mean diameter, and divi- 
ded by the sun's mean diameter : subtract the result from the 
earth's corrected diameter (stfH), and the remainder is the diam- 
eter of the shadow; which is reduced to minutes as before. 

A- 

The method employed in this process for finding the diameter of the 
earth's shadow upon the moon's mean orbit may be explained by the 
aid of the following figure (Fig. 20). 

As in the last figure, let £ represent the earth's place, S and M points 
in the mean orbits of the sun and moon, and M' the moon's actual 
place. Let t u be the sun's corrected diameter, or the part of his mean 
orbit which his disk at its actual distance covers, ascertained as directed 
in the preceding passage, and let FG be the earth's diameter. Through 

Fig. 20. 




F and G draw vF/and w Gg parallel to S M, and also t¥k and uOk : 
then h h will be the diameter of the shadow where the moon actually 
enters it The value of hk evidently equals fg (or FG)— (Jh^gh) ; 
and the value of fh+g k may be found by the proportion 

Fv (or ES) : tv+wu (or <*-FG) ::F/(or EM') :fh+gk 

'&ri the Hindu system provides no method of measuring the angular 
v^oe of quantities at the distance E M', nor does it ascertain the value 
of EM ; itself; and as,, in the last process, the diameter of the moon 



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274 HL Burgess, etc, [iv. 5- 

was reduced, for measurement* to its value at the distance £ M', so, to 
be made commensurate with it, all the data of this process must be 
similarly modified. That is to say, the proportion 

EM': EM ::/>:// 
-^•substituting, as before, the ratio of the moon's mean to her true 
motion for that of EM' to EM — gives fg'> which the text calls the 
*uci : the word means literally u needle, pyramid ; we do not see pre- 
cisely how it comes to be employed to designate the quantity f g'y and 
have translated it, for lack of a better term, and in analogy with the 
language of the text respecting the diameters of the sun and moon, 
" corrected diameter of the earth." It is also evident that 

EM':fk+gk::EM:fh'+g'k' 

hence, substituting the latter of these ratios for the former in our first 
proportion, and inverting the middle terms, we have 

ES : E M : : <u-FG :fh'+g'k' 
Once more, now, we have a substitution of ratios, E S : E M being re- 
placed by the ratio of the sun's mean diameter to that of the moon. 
In this there is a slight inaccuracy. The substitution proceeds upon the 
assumption that the mean apparent values of the diameters of the sun 
and moon are precisely equal, in which case, of course, their absolute^ 
diameters would be as their distances ; but we have seen, in the note to 
the first verse of this chapter, that the moon's mean angular diameter is 
made a little less than the sun's, the former being 32', the latter 32' 24".8. 
The error is evidently neglected as being too small to impair sensibly the 
correctness of the result obtained : it is not easy to see, however, why we 
do not have the ratio of the mean distances represented here, as in verses 
2 and 3, by that of the orbits, or by that of the revolutions in an Age 
taken inversely. The substitution being made, we have the final propor- 
tion on which the rule in the text is based, viz^ the sun's mean diameter 
is to the moon's mean diameter as the excess of the suu's corrected 
diameter over the actual diameter of the earth is to a quantity which, 
being subtracted from the suet, or corrected diameter of the earth, leaves 
as a remainder the diameter of the shadow as projected upon the moon's 
mean orbit : it is expressed in yojanas, but is reduced to minutes, as 
before, by dividing by fifteen. The earth's penumbra is not taken into 
account in the Hindu process of calculation of an eclipse. 

The lines fg,fg\ etc., are treated here as if they were straight lines, 
instead of arcs of the moon's orbit : but the inaccuracy never comes to 
be of any account practically, since the value of these lines always falls 
inside of the limits within which the Hindu methods of calculation 
recognize no difference between an arc and its sine. 

6. The earth's shadow is distant half the signs from the sun : 
when the longitude of the moon's node is the same with that of 
the shadow, or with that of the sun, or when it is a few degrees 
greater or less, there will be an eclipse. . . 

To the specifications of this verse we need to add, of course, " at the 
time of conjunction or of opposition.*' 



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xv. 8.] S&rya-JMdli&nia. 273 

It will be noticed that no attempt is mode hereto define the lunar 
and solar ecliptic limits or the distances from the moon's node withm 
which eclipses are possible. . Those limits, are,, for the moon,, nearly 12° * r 
for the sun, more than 17°. 

The word used to designate "eclipse," grahana, means literally 
"sci2ure" : it, with other kindred terms, to be noticed later, exhibits the 
influence of the primitive theory of eclipses, as seizures of the heavenly 
bodies by the monster Rahu. In verses 17 and 19, below, instead of 
grahana we have graha, another derivative from the same root grah or 
grabh, u grasp, seize.' 1 Elsewhere graka never occurs except as signifying 
"planet," and it is the only word which the Surya-Sidananta employs, 
with that signification! as so used, it is an active instead of a passive 
derivative, meaning " seizer " and its application to the planets is due 
to the astrological conception of them, as powers which u lay hold upon" 
the fates of men with their supernatural influences. 

7. The longitudes of the sun and moon, at the moment of 
the end of the day of new moon (amdt>dsyd) } are equal, in signs, 
etc. ; at the end of the day of full moon {pammamdA) they are 
equal in degrees, etc., at a distance of half the signs. 

8. When diminished or increased by the proper equation of : 
motion for the time, past or to come, of opposition or conjunc- 
tion, they are made to agree, to minutes: the place of the node 
at the same time is treated in the contrary manner. 

The very general directions and explanations contained in verses 6, 7, 
and 9 seem out of place hero in the middle of the chapter, and would 
have more properly constituted its introduction. The process prescribed ; 
in verse 8, also, which has for its object the determination of the longi- 
tuded of the sun, moon, and moon's node, at the moment of opposition 
or conjunction, ought no less, it would appear, to precede the ascertain- 
ment of the true motions, and of the measures of the disks and shadow, 
already explained. Verse 8, indeed, by the lack of connection in which 
it stands, and by the obscurity of its language, furnishes a striking in- 
stance of the want of precision and intelligibility so often characteristic : 
of the treatise. The subject of the verse, which requires to be supplied, 
is, " the longitudes -of the sun and moon at the instant of midnight next - 
preceding or following the given opposition or conjunction" ; that being - 
tbfe time for which the true longitudes and motions are first calculated, 
in order to test the question of the probability of an eclipse. If there 
appears to be such a probability, the next step is to ascertain the inter- 
val between midnight and the moment of opposition or conjunction, 
past or to come : this is done by the method taught in ii. 66, or by some 
other analogous process : the instant of the occurrence of opposition or 
conjunction, in local time, counted from sunrise of the place of observa- 
tion, must also be determined, by ascertaining the interval between mean 
and apparent midnight (ii. 46), the length of the complete day (ii. 59), 
and of its parts (ii. 60-63), etc. ; the whole process is sufficiently illus- 
trated by the two examples of the calculation of eclipses given in the 
Appendix, When we have thus found the interval between miilnight 



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276 fl Burgess, etc., [iv. 8- 

and the moment of opposition or conjunction, verse 8 teaches us how to 
ascertain the true longitudes for that moment : it is by calculating — in 
the manner taught in i. 67, but with the true daily motions — the 
amount of motion of the sun, moon, and node during the interval, and 
applying it as a corrective equation to the longitude of each at mid- 
night, subtracting in the case of the sun and moon, and adding in the 
case of the node, if the moment was then already past ; and the con- 
trary, if it was still to come. Then, if the process has been correctly 
performed, the longitudes of the sun and moon will be found to corres- 
pond, in the manner required by verse 7. 

For the days of new and full moon, and their appellations, see tho 
note to ii. 66, above. The technical expression employed here, as in 
one or two other passages, to designate the " moment of opposition or 
conjunction" is parvanddyas, "nadis of the parvan," or "time of the 
parvan in nadis, etc. :" parvan means literally "knob, joint," and is fre- 
quently applied, as in this term, to denote a conjuncture, the moment 
that distinguishes and separates two intervals, and especially one that is 
of prominence and importance. 

9. The moon is the eclipser of the sun, coming to stand under- 
neath it, like a oloud: the moon, moving eastward, enters the 
earth's shadow, and the latter becomes its eclipser. 

The names given to the eclipsed and eclipsing bodies are either chAdya 
and, as here, ch&daka, "the body to be obscured" and "the obscurer," 
or grdkya and grdhaka, "the body to be seized" and "the seiser." 
The latter terms are akin with grahana and graha, spoken of above 
(note to v. 6), and represent the ancient theory of the phenomena, while 
the others are derived from their modern and scientific explanation, as 
given in this verse. 

10. Subtract the moon's latitude at the time of opposition or 
conjunction from half the sum of the measures of the eclipsed 
and eclipsing bodies : whatever the remainder is, that is said to 
be the amount obscured. 

11. When that remainder is greater than the eclipsed body, 
the eclipse is total ; when the contrary, it is partial ; when the 
latitude is greater than the half sum, there takes place no obscu- 
ration^&a). ,. 

It is; Sufficiently evident that, when, at the moment of opposition,. th,e 
moon's latitude— -which is the distance of her centre from the ecliptic, . 
where is the centre of the shadow— is equal to the sum of the radii of 
her disk and of the shadow, the disk and the shadow will just touch one 
another ; and that, on the other hand, the moon will, at the moment of 
opposition, be so far immersed in the shadow as her latitude is less than 
the sum of the radii : and so in like manner for the sun, with due allow- 
ance for parallax. The Hindu mode of reckoning the amount eclipsed 
is not by digits, or twelfths of the diameter of the eclipsed body, which 
method we nave inherited from the Greeks, but by minutes. 



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iv. 13.] Stlrya-Siddhdnta. 277 

The word gr&sa, used in verse 1 1 for obscuration or eclipse, means 
literally " eating, devouring," and so speaks more distinctly than any 
other term we have had of the old theory of the physical cause of 
eclipses, 

12. Divide by two the sura and difference respectively of the 
eclipsed and eclipsing bodies: from the square of each of the 
resulting quantities subtract the square of the latitude, and take 
the square roots of the two remainders. 

13. These, multiplied by sixty and divided by the difference 
of the daily motions of the sun and moon, give, in nfidls, etc., 
half the duration (sthiti) of the eclipse, and half the time of total 
obscuration. 

Thee* rules for finding the intervals of time between the moment 
of opposition or conjunction in longitude, which is regarded as the 
middle of the eclipse, and the moments of first and last contact, and, in 
a total eclipse, of the beginning and end of total obscuration, may be 
illustrated by help of the annexed figure (Fig. 21). 

Let E C L represent the ecliptic, the point G being the centre of the 
shadow, and let C D be the moon's latitude at the moment of opposi- 




tion; which, for the present, we will suppose to remain unchanged 
through the whole continuance of the eclipse. It is evident that the 
first contact of the moon with the shadow will take place when, in the 
triangle CAM, AC equals the moon's distance in longitude from the 
centre of the shadow, A M her latitude, and C M the sum of her radius 
and that of the shadow. In like manner, the moon will disappear en- 
tirely within the shadow when B C equals her distance in longitude from 
the centre of the shadow, B N her latitude, and C N the difference of 
the two radii. Upon subtracting, then, the square of A M or B N from 
those of C M and C N respectively, and taking the square roots of the 
remainders, we shall have the values of A C and B C in minutes. These 
may be reduced to time by the following proportion : as the excess at 
vol. vi. 36 



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278 E. Burgess, etc,, [iv. 13- 

the given time of the moon's true motion in a day over that of the sun 
is to a day, or sixty n&dls, so are A G and B C, the amounts which the 
moon has to gain in longitude upon the sun between the moments of 
contact and immersion respectively and the moment of opposition, to 
the corresponding intervals of time. 

But the process, as thus conducted, involves a serious error : the 
moon's latitude, instead of remaining constant during the eclipse, is con- 
stantly and sensibly changing. Thus, in the figure above, of which the 
conditions are those found by the Hindu processes for the eclipse of Feb. 
6th, 1860, the moon's path, instead of being upon the line HK, parallel 
to the ecliptic, is really upon Q R. The object of the process next 
taught is to get rid of this error. 

14. Multiply the daily motions by the half-duration, in nadia, 
and divide by sixty : the result, in minutes, subtract for the time 
of contact (pragraka), and add for that of separation (moksha), 
respectively ; 

15. By the latitudes thence derived, the half-duration, and 
likewise the half-time of total obscuration, are to be calculated 
anew, and the process repeated. In the case of the node, the 
proper correction, in minutes, etc., is to be applied in the con- 
trary manner. 

This method of eliminating the error involved in the supposition of 
a constant latitude, and of obtaining another and more accurate deter- 
mination of the intervals between the moment of opposition and those 
of first and last contact, and of immersion and emergence, is by a series 
of successive approximations. For instance : A C, as already determined, 
being assumed as the interval between opposition and first contact^ a 
new calculation of the moon's longitude is made for the moment A, and, 
with this and the sum of the radii, a new value is found for A C. But 
now, as the position of A is changed, the former determination of its 
latitude is vitiated and must be made anew, and made to furnish anew 
a corrected value of A C ; and so on, until the position of A is fixed 
with the degree of accuracy required. The process must be conducted 
separately, of course, for each of the four quantities affected ; since, where 
latitude is increasing, as in the case illustrated, the true values of A C 
and B C will be greater than their mean values, while G C and F C, the 
true intervals in the after part of the eclipse, will be less than A C and 
B C : and the contrary when latitude is decreasing. 

We have illustrated these processes by reference only to a lunar 
eclipse : their application to the conditions of a solar eclipse requires 
the introduction of another element, that of the parallax, and will be 
explained in the notes upon the next chapter. 

The first contact of the eclipsed and eclipsing bodies is styled in this 
passage pragraha, " seizing upon, laying hold of;" elsewhere it is also 
called gr&sa, " devouring," and sparpa, a touching :" the last contact, or 
separation, is named moksha, u release, letting go." The whole duration 
of the eclipse, from contact to separation, is the athiti, " stay, continu- 
ance ; " total obscuration is vimaraa, " crushing out, entire destruction." 



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iv. 21.] SCtrya-Siddhdnta. 279 

16. The middle of the eclipse is to be regarded as occurring 
at the true close of the lunar day : if from that time the time of 
half-duration be subtracted, the moment of contact (grdsa) is 
found ; if the same be added, the moment of separation. 

17. In like manner also, if from and to it there be subtracted 
and added, in the case of a total eclipse, the half-time of total 
obscuration, the results will be the moments called those of im- 
mersion and emergence. 

The instant of true opposition, or of apparent conjunction (see below, 
under ch. v. 9), in longitude, of the sun and moon, is to be taken as the 
middle of the eclipse, even though, owing to the motion of the moon in 
latitude, and also, in a solar eclipse, to parallax, that instant is not mid- 
way between those of contact and separation, or of immersion and 
emergence. To ascertain the moment of local time of each of these 
phases of the eclipse, we subtract and add, from and to the local time 
of opposition or conjunction, the true intervals found by the processes 
described in verses 12 to 15. 

The total disappearance of the eclipsed body within, or behind, the 
eclipsing body, is called nimilana, literally the " closure of the eyelids, 
as in winking : " its first commencement of reappearance is styled unmt- 
lana, u parting of the eyelids, peeping." We translate the terms by 
"immersion" and "emergence" respectively. 

18. If from half the duration of the eclipse any given interval 
be subtracted, and the remainder multiplied by the difference of 
the daily motions of the sun and moon, and divided by sixty, the 
result will be the perpendicular (koti) in minutes. 

19. In the case of an eclipse (graha) of the sun, the perpen- 
dicular in minutes is to be multiplied by the mean half-duration, 
and divided by the true (sphuta) half-duration, to give the true 
perpendicular in minutes. 

20. The latitude is the base (bhuja) : the square root of the 
sum of their squares is the hypothenuse (grava) : subtract this 
from half the sum of the measures, and the remainder is the 
amount of obsct#ation {grdsa) at the given time. 

21. If tJ&at time be after the middle of the eclipse, subtract 
the interval from the half-duration on the side of separation, and 
treat the remainder as before : the result is the amount remaining 
obscured on the side of separation. 

The object of the process taught in this passage is to determine the 
amount of obscuration of the eclipsed body at any given moment during 
the continuance of the eclipse. It, as well as that prescribed in the 
following passage, is a variation of that which forms the subject of verses 
12 and 13 above, being founded, like the latter, upon a consideration of 
the right-angled triangle formed by the line joining the centres of the 
eclipsed and eclipsing bodies as hypothenuse, the difference of their 
longitudes as perpendicular, and the moon's latitude as base. And 
whereas, in the former problem, we had the base and hypothenuse given 



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280 R Burgess, etc., [W. 21- 

to find the perpendicular, her© we. have the base and perpendiculaxgiven 
to find the hypothenuse. The perpendicular is furnished us ju time, 
and the rule supposes it to be stated in the form of the interval between 
the given moment and that of contact or of separation : a form to 
which, of course, it may readily be reduced from . any other mode of 
statement The interval of time is reduced to its equivalent as differ- 
ence of longitude by a proportion the reverse of that given in verse 13, 
by which difference of longitude was converted into time ; the moon's 
latitude is then calculated ; from the two the hypothenuse is deduced ; 
and the comparison of this with the sum of the radii gives the measure 
of the amount of obscuration. 

Verse 21 seems altogether superfluous : it merely states the method of 
proceeding in case the time given falls anywhere between the middle and 
the end of the eclipse, as if the specifications of the preceding verses ap- 
plied only to a time occurring before the middle : whereas they are gen- 
eral in their character, and include the former case no less than the latter. 

When the eclipse is one of the sun, allowance needs to be made for 
the variation of parallax during its continuance } this is done by the 
process described in verse 19, of which the explanation will be given in 
the notes to the next chapter (vv. 14-17). 

In verse 20, for the first and only time, we have latitude called kthepi, 
instead of vikskepa, as elsewhere. In the same verse, the term employed 
for "hypothenuse" is frava, "hearing, organ of hearing;" this, as well 
as the kindred pravana, which is also once or twice employed, is a syno- 
nym of the ordinary term karna, which means literally " ear." It is 
difficult to see upon what conception their employment in this significa- 
tion is founded. 

22. From half the sum of the eclipsed and eclipsing bodies 
subtract any given amount of obscuration, in minutes: from the 
square of the remainder subtract the square of the latitude at 
the time, and take the square root of their difference. 

23. The result is the perpendicular (Jcbti) in minutes — which, 
in an eclipse of the sun, is to be multiplied by the true, and 
divided bv the mean, half-duration — and this, converted intp 
time by the same manner as when finding th% duration of the 
eclipse, gives the time of the given amount of obscuration (grdsa). 

The conditions of this problem are precisely the same with those of 
the problem stated above, in verses 12-15, excepting that here, instead 
of requiring the instant of time when obscuration commences, or becomes 
total, we desire to know when it will be of a certain given amount. 
The solution must be, as before, by a succession of approximative steps, 
since, the time not being fixed, the corresponding latitude of the moon 
cannot be otherwise determined. 

24. Multiply the sine of the hour-angle (nata) by the sine of 
the latitude (aicsha), and divide by radius : the arc correspond- 
ing to the result is the degrees of deflection (valandnfds), which 
are north and south in the eastern and western hemispheres 
(kapdla) respectively. 



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iv. 26.] S&rya-Siddhdnta. 281 

. 25. From the position of the eclipsed body increased by three 
-wgos calculate toe de^ees of declination : addthemto the de- 
grees of deflection, if of like direction ; take their difference, if of 
aiffereot direction: the corresponding sine is the deflection (pa- 
land) — in digits, when divided by seventy. 

This process requires to be performed only when it is desired to pro- 
ject an eclipse. In making a projection according to the Hindu method, 
as wiU be seen in connection with the sixth chapter, the eclipsed body 
is represented as fixed in the centre of the figure, with a north anil 
south line, and an east and west line, drawn through it The absolute 
position of these lines upon the disk of the eclipsed body is, of course, 
all the time changing : but the change is, in the case of the sun, not 
observable, and in the case of the moon it is disregarded : the Surya- 
Siddh&nta takes no notice of the figure visible in the moon's face as 
determining feny fixed and natural directions upon her disk. It is de- 
sired to represent to the eye, by the figure drawn, where, with reference 
to the north, south, east, and west points of the moment, the contact, im- 
mersion, emergence, separation, or other phases of the eclipse, will take 
place. In order to this, it is necessary to know what is, at each given 
moment, the direction of the ecliptic, in which the motions of both 
eclipsed and eclipsing bodies are made. The east and west direction is 
represented by a small circle drawn through the eclipsed body, parallel 
to the prime vertical ; the north and south direction, by a great circle 
passing through the body and through the north and south points of 
the horizon : and the direction of the ecliptic is determined by ascer- 
certaining the angular p. 22 

amount of its deflection s ' 

from the small east and 
west circle at the point 
occupied by the eclipsed 
body. Thus, in the an- 
nexed figure (Pig. 22), 
if M be the place of 
the eclipsed body upon 
the ecliptic, C L, and if 
EW be the small east 
and west circle drawn 
through M parallel with 
E' Z, the prime vertical, then the deflection will be the angle made at M 
by C M and E M, which is equal to P' M N, the an^le made by perpen- 
diculars to the two circles drawn from their respective poles. In order 
to find the value of this angle, a double process is adopted : first, the 
angle made at M by the two small circles E M and D M, which is equiv- 
alent toPMN, is approximately determined : as this depends for its 
amount upon the observer's latitude, being nothing in a right sphere, it 
is called by the commentary dkska valana, "the deflection due to lati- 
tude :" the text calls it simply valan&nf&s, " degrees of deflection," since 
it does not, like the net result of the whole operation, require to be ex- 
pressed in terms of its sine. Next, the angle made at M by the ecliptic, 




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282 



U. Burgess, etc., 



[iv. 25. 



Fig. 28. 



C L, and the circle of daily revolution, D R, *which angle is equal to 
P M P', is also measured : this the commentary calls dyana valana, "the 
deflection due to the deviation of the ecliptic from the equator ;" the 
text has no special name for it. The sum of these two results, or their 
difference, as the case may be, is the valana, or the deflection of the 
ecliptic from the small east and west circle at M, or the angle P' M N. 

In explaining the method and value of these processes, we will com- 
mence with the second one, or with that by which P M P', the at/ana 
valana, is found. In the following figure (Fig. 23), let O Q be the 
equator, and M L the ecliptic, P and P' being their respective poles. 
Let M be the point at which the amount of deflection of M L from the 
circle of diurnal revolution, D R, is sought. Let M L equal a quadrant ; 
draw P' L, cutting the equator at Q ; 
as also P L, cutting it at B ; then draw 
PM and QM. Now P'ML is a tri- 
quadrantal triangle, and hence M Q is 
a quadrant; and therefore Q is a pole 
of the circle POM, and Q O is also a 
quadrant, and Q M O is a right angle. 
But D B also makes right angles at M 
with PM; hence QM and DR are 
tangents to one another at M, and the 
spherical angle QML is equal to that 
which the ecliptic makes at M with the 
circle of declination, or to P M P' : and 
QML is measured by QL. The rule 
given in the text produces a result which 
is a near approach to this, although not 
entirely accordant with it excepting at the solstice and equinox, the 
points where the deflection is greatest and where it is nothing. We 
are directed to reckon forward a quadrant from the position of the 
eclipsed body — that is, from M to L, in the figure — and then to calcu- 
late the declination at that point, which will be the amount of deflection. 
But the declination at L is BL, and since LBQ is a right-angled 
triangle, having a right angle at B, and since L Q and L B are always 
less than quadrants, L B must be less than L Q. The difference between 
them, however, can never be of more than trifling amount ; for, as the 
angle QLB increases, QL diminishes ; and the contrary. 

In order to show how the Hindus have arrived at a determination of 
this jjart of the deflection so nearly correct, and yet not quite correct, 
we Will cite the commentator's explanation of the process. He says: 
"The 'east' (pr&cfy of the equator [i.e.; apparently, the point of the 
equator eastward toward which the small circle must be considered as 
pointing at M] is a point 90° distant from that where a circle' drawn 
from the pole (dkruva) through the planet cuts the equator ? that is to 
say, it is the f>oint Q (Fig. 28), a quadrant from O : u and the interval 
by which this is separated from the ' east' of the ecliptic at 00° from the 
planet, that is the dyana valanaP This is entirely correct, and would 
give us QL, the true measure of the deflection. But the commentator 
goes on farther to say that since this interval, when the planet is at the 




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iv. 25.] S&rya-Siddh&nta. 283 

solstice, is nothing, and when at the equinox is equal to the greatest 
declination, it is therefore always equal to the declination at a quadrant's 
distance from the planet This is, as we have seen, a false conclusion, 
and leads to an erroneous result : whether they who made the rule were 
aware of this, but deemed the process a convenient one, and its result a 
Mifficiently near approximation to the truth, we will not venture to say. 

The other part of the operation, to determine the amount of deflec- 
tion of the circle of declination from the east and west small circle, is 
considerably more difficult, and the Hindu process correspondingly 
defective. We will first present the explanation of it which the com- 
mentator gives. He states the problem thus : " by whatever interval 
the directions of the equator are deflected from directions correspond- 
ing to those of the prime vertical, northward or southward, that is the 
deflection due to latitude (dksha valana). Now then : if a movable 
circle be drawn through the pole of the prime vertical (soma) and the 
point occupied by the planet [i. e., the circle N M S, Fig. 22], then the 
interval of the * easts,' at the distance of a quadrant upon each of the 
two circles, the equator and the prime vertical, from the points where 
they are respectively cut by that circle [i. e., from T and V] will be the 
deflection. . . . Now when the planet is at the horizon [as at D, referred 
to £'], then that interval is equal to the latitude [Z Q] ; when the planet 
is upon the meridian (y&myottaravrtta, " south and north circle") [i e., 
when it is at R, referred to Q and Z], there is no interval fas at E'1. 
Hence, by the following proportion — with a sine of the hour-angle 
which is equal to radius the sine of deflection for latitude is equal to 
the sine of latitude ; then with any given sine of the hour-an^le what 
is it? — a sine of latitude is found, of which the arc is the required de- 
flection for latitude." This is, in tue Hindu form of statement, the 
proportion represented by the rule in verse 24, viz. R : sin hit : : sin 
hour-angle : sin deflection. 

It seems to us very questionable, at least, whether the Hindus had 
any more rigorous demonstration than this of the process they adopted, 
or knew wherein lay the inaccuracies of the latter. These we will now 
proceed to point out In the first place, instead of measuring the angle 
made at the point in question, M, by the two small circles, the east and 
west circle and that of daily revolution*— which would be the angle 
P M N — they refer the body to the equator by a circle passing through 
the north and south points of the horizon, and measure the deflection 
of the equator from a small east and west circle at its intersection with 
that oircfe — which is the angle P T N. Or, if we suppose that, in the 
process formerly explained, no regard was had to the circle of dairy 
revolution, D R, the intention being to measure the difference in direc- 
tion of the ecliptic at M and the equator at O, then the two parts of 
the process are inconsistent in this, that the one takes as its equatorial 
point of measurement 0, and the other T, at which two points the 
direction of the equator is different But neither is the value of P T N 
correctly found. For, in the spherical triangle P N T, to find the angle 
at T, we should make the proportion 

sin PT (or R) : sin PN : : sin PNT : sin PTN 
But, as the third term in this proportion, the Hindus introduce the sine 



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284 R Burgess, etc*, [iv. 2S-" 

of the hour-angle, ZPM or MPN, although with a certain modifica- 
tion which the commentary prescribes, and which makes of it some- 
thing very near the angle T P N. The text says simply natajya, " the 
sine of the hour-angle 7 ' (for nata y see notes to iii. 84-36, and 14-16), 
but the commentary specifies that, to find the desired angle in degrees, 
we must multiply the hour-angle in time by 90, and divide by the haj^ 
day of the planet This is equivalent to making a quadrant of that 
part of the circle of diurnal revolution which is between the horizon 
and the meridian, or to measuring distances upon DR as if they were 
proportional parte of E' Q. To make the Hindu process correct, the 

Eroduct of this modification should be the angle P N T, with which, 
owever, it only coincides at the horizon, where both TPN and TNP 
become right-angles, and at the meridian, where both are reduced to 
nullity. The error is closely analogous to that involved in the former 
process, and is of slight account when latitude is small, as is also the 
error in substituting T for O or M when neither the latitude nor the 
declination is great 

The direction of the ecliptic deflection (dyana valana) is the same/ 
evidently, with that of the declination a quadrant eastward from the 
point in question ; thus, in the case illustrated by the figjuie, i^s f souths 
The direction of the equatorial deflection (dksha valana) depends upon 
the .position of the point considered, with reference to the- meridian*, 
being — in northern latitudes,' which alone the Hindu system SQBtoHtt 
plates->-north when that point is east of the meridian, and. south when 
west of It," as specified In verse 24: since, for instance, fi' being the 
east point of the horizon, the equator at any point between E' and Q 
point*, eastward, toward a point north of the prim* vertical -Inlhe 
case for which the figure is drawn, then, the difference tff the twowouii 
be the finally "resulting deflection. Since, in making "the projection^ 
the eclipse, it is laid off as a straight line (see the iltsstration given inT 
connection with chapter viV it must be reduced to its value as a siueV 
and moreover, since it is laid down in a circle of which the radius is' 
49 digits (see below, vi 2), or in: which one. digit equals' faV-^fon 
3488'-r49 — 70', nearly — that sine is reduced to its value in digits by 
dividing it by 70. 

The general subject of this ' passage, tfce~determination of directions 
during an eclipse, for the purpose of establishing the positions, upon the 
disk of the eclipsed body, <of the, points of -contact, immersion, emerg- 
ence, and separation, also engaged the attention of the Greeks ; Ptolemy 
devotes to it the eleventh and twelfth chapters of the sixth book of his 
Syntaxis : his representation of directions, however, and consequently 
his method of calculation also, are different from those here cxpeocti .*^ 

'. 26. To the altitude in time (unncUa) add a day and a half* and 
divide by a half-day; by the quotient divide the latitudes. and 
the disks; the results are the measures of those qnantttie^in 
digits (cmgula). . .> 

By this process due account is taken, in the projection of an eclipse, 
of the apparent increase in magnitude of the heavenly bodies when 
near the horizon. The theory lying at the foundation of the rule is this % 



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▼. 1.] Surya-SiddJxdnla. 285 

that three minutes of arc at the horizon, and four at the zenith, aro 
equal to a digit, the difference between the two, or the excess above 
three minutes of the equivalent of a digit at the zenith, being one 
minute. To ascertain, then, what will be, at any given altitude, the 
excess above three minutes of the equivalent of a digit, we ought prop- 
erly, according to the commentary, to make the proportion 

R : 1 ' : : sin altitude : corresp. excess 
Since, however, it would be a long and tedious process to find the alti- 
tude and its sine, another and approximative proportion is substituted 
for this " by the blessed Sun," as the commentary phrases it, " through 
compassion for mankind, and out of regard to the very slight difference 
between the two." It is assumed that the scale of four minutes to the 
digit will be always the true one at the noon of the planet in question, 
or whenever it crosses the meridian, although not at the zenith ; and so 
likewise, that the relation of the altitude to 90° may be measured by 
that of the time since rising or until setting (unnata — see above, iii. 
37-39) to a half-day. Hence the proportion hecomes 

half-day : 1' : : altitude in time : corresp. excess 
and the excess of the digital equivalent above 8' equals at ! "l in ^. 

Adding, now, the three minutes, and bringing them into the fractional 
expression, we have 

equiv. of digit in minutes at given time a ^1^^^ Mf dav » 

The title of the fourth chapter is candragmhan&dkik&ra, " chapter of 
lunar eclipses," as that of the fifth is storyagrahtindkdhik&ra y " chapter of 
solar eclipses." In truth, however, the processes and explanations of 
this chapter apply not less to solar than to lunar eclipses, while the next 
treats only of parallax, as entering into the calculation of a solar eclipse. 
We have taken the liberty, therefore, of modifying accordingly the 
headings which we have prefixed to the chapters. 



CHAPTER V. 

OF PARALLAX IN A SOLAR ECLIFS& 

Comnmt— 1, when there is no parallax in longitude, or no parallax in latitude; 
I, causes of parallax; 3, to find the orient-sine ; 4-5, the meridian-sine ; 5-7, and 
fbe sines of ecliptic senitiMttstance and altitude ; 7*6, to find the amount, in time, 
of the parallax m longitude ; 9, its application in determining the moment of 
apparent conjunction; 10-11, to find the amount, in arc, of the parallax in lati- 
tude; 1S-13, its application in calculating an eclipte; 14-17, application of the 
parallax in longitude in determining the moments of contact, of separation, etc 

1. When the sun's place is coincident with the meridian 
ecliptiopoint (madhyalagna), there takes place no parallax in 
vol. vi. 37 



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286 



& Burgess 1 etc., , 



[v. 1. 



longitude (liarija) : farther, when terrestrial latitude (aksha) and 
north declination of the meridian ecliptic-point (madhyabha) are 
the same, there takes place no parallax in latitude (avanati). 

The latter of these specifications is entirely accurate : when the north 
declination of that point of the ecliptic which is at the moment upon 
the meridian (madhyalagna ; see iii. 49) is equal to the observer's lati- 
tude — regarded by the Hindus as always north — the ecliptic itself 
passes through the zenith, and becomes a vertical circle ; of course, then, 
the effect of parallax would be only to depress the body in that circle, 
not to throw it out of it. The other is less exact : when the sun is 
upon the meridian, there is, indeed, no parallax in right ascension, but 
there is parallax in longitude, unless the ecliptic is also bisected by the 
meridian. Here, as below, in verses 8 and 9, the text commits the 
inaccuracy of substituting the meridian ecliptic-point (L in Fig. 26) for 
the central or highest point of the ecliptic (13 in the same figure). The 
latter point, although we are taught below (vv. 5-7) to calculate the sine 
and cosine of its zenith-distauce, is not once distinctly mentioned in the 
text ; the commentary calls it tribh<malagna y " the orient ecliptic-point 
(lagna — see above, iii. 46-48 : it is the point C in Fig. 26) less three 
signs." The commentary points out this inaccuracy on the part of the text, 
in order to illustrate the Hindu method of looking at the subject of 
parallax, we make the following citation from the general exposition of 
it given by the commentator under this verse : " At the end of the day 
of new moon (am&v&syd) the sun and moon have the same longitude ; 
if, now, the moon has no latitude, then a line drawn from the earth's 

centre [C in the accompanying 
^' 24 figure] to the sun's place [S] just 

touches the moon [M] : hence, 
at the centre, the moon becomes 
an eclipsing, and the sun an 
eclipsod, body. Since, however, 
men are not at the earth's centre, 
(garbha, " womb ") but upon the 
earth's surface {prshthay " back"), 
a line drawn from the earth's 
surface [B] up to the sun does 
not just touch the moon ; but it 
cuts the moon's sphere above the 
point occupied by the moon [at 
m], and when the moon arrives 
at this point, then is she at the 
earth's surface the eclipser of 
the sun. But when the sun is at 
the zenith {khamadhya, "mid- 
heaven"), then the lines drawn up to the sun from the earth's centre 
and surface, being one and the same, touch the moon, and so the moon 
becomes an eclipsing body at the end of the day of new moon. Hence, 
too, the interval [M m] of the lines from the earth's centre and surface 
is the parallax (tamfatna)" 




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v. l.] S&rya-Siddhdnta. 287 

It is evident from this explication how far the Hindu view of parallax 
is coincident with our own. The principle is the same, but its applica- 
tion is somewhat different. Instead of taking the parallax absolutely, 
determining that for the sun, which is B S 0, and that for the moon, 
which is B M C, the Hindus look at the subject practically, as it must be 
taken account of in the calculation of an eclipse, and calculate only the 
difference of the two parallaxes, which is m B M, or, what is virtually 
\\ie same thing, M C m. The Surya-Siddhanta, however, as we shall see 
iereafter more plainly, takes no account of any case in which the lino 
iv S would not pass through M, that is to say, the moon's latitude is 
neglected, and her parallax calculated as if she were in the ecliptic. 

We cite farther from the commentary, in illustration of the resolution 
of the parallax into parallax in longitude and parallax in latitude. 

u Now by how many degrees, measured on the moon's sphere (golaY 
the line drawn from the earth's surface up to the sun cuts the moon s 
vertical circle (drgvrtia) above the point occupied by the moon — this is, 
when the vertical circle and the ecliptic coincide, the moon's parallax in 
longitude (lambana). But when the ecliptic deviates from a vertical 
circle, then, to the point where the line from the earth's surface cuts the 
moon's sphere on the moon's vertical circle above the moon [i. e., to m, 
p 25 Fig. 25], draw through the pole 

of the ecliptic (kadamba) a cir- 
cle [P'mn] north and south to 
the ecliptic on the moon's sphere 
[M n'] : and then the east and 
west interval [Mn'] on the eclip- 
tic between the point occupied 
by the moon [MJ and the point 
where the circle as drawn cuts 
the ecliptic on the moon's sphere 
[n'] is the moon's true (sphuta) 
parallax in longitude, in minutes, and is the perpendicular (koti). And 
since the moon moves along with the ecliptic, the north and south inter- 
val, upon the circlo we have drawn, between the ecliptic and the vertical 
circle [m n'] is, in minutes, the parallax in latitude (nati) ; which is the 
base (bhuja). The interval, in minutes, on the vertical circle [ZA], 
between the lines from the earth's centre and surface [m M], is the ver- 
tical parallax (drglambana), and the hypothenuse." 

The conception here presented, it will be noticed, is that the moon's 
path, or the " ecliptic on the moon's sphere," is depressed away from 
C L, which might be called the " ecliptic on the sun's sphere," to an 
amount measured as latitude by m ri, and as longitude by n' M. To 
onr apprehension, m»M, rather than mn'M, would be the triangle of 
resolution : the two are virtually equal. 

The commentary then goes on farther to explain that when the ver- 
tical circle ana the secondary to the ecliptic coincide, the parallax in 
longitude disappears, the whole vertical parallax becoming parallax in 
latitude : and again, when the vertical circle and the ecliptic coincide, 
the parallax in latitude disappears, the whole vertical parallax becoming 
parallax in longitude. 




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288 E % Burgess, etc.> [t. l-> 

. The term uniformly employed by the commentary, and. more usually 
by the text, to express parallax in longitude, namely lambana, is front. 
the same root which we have already more than once had occasion to 
notice (Bee above, under i. 25, 60), and means literally u hanging down-, 
ward." In this verse, as once or twice later (vv. 14, 16), the text uses 
karija, which the commentary ex plaint as equivalent to ktkitija, "pro- 
duced by the earth:" this does not seem very plausible, but we have! 
nothing better to suggest. For parallax in latitude the text present* 
only the term avanati, "bending downward, depression:" the commen* 
tary always substitutes for it na/t, which has nearly the same sense, and 
is the customary modern term. 

2. How parallax in latitude arises by reason of the difference 
of place (dtqa) and time (k&la\ and also parallax; in longitude 
(Utndxina) from direction (dig) eastward or the contrary — that i* 
jiow to be explained. 

This distribution of the three elements of direction, place, and time, 
as causes respectively of parallax in longitude and in latitude, is some- 
what arbitrary. The verse is to be taken, however, rather as a general 
introduction to the subject of the chapter, than as a systematic state' 
ment of the causes of parallax. 

3. Calculate, by the equivalents in oblique ascension (udayd- 
savas) of the observer's place, the orient ecliptic-point (lagna) far 
the moment of conjunction (parvavinddyas) : multiply the sine 
of its longitude by the sine of greatest declination, and divide 
by the sine of co*latitude (lamoa) : the result is the quantity 
known as the orient-sine (udaya). 

The object of this first step in the rather tedious operation of calcu- 
lating the parallax is to find for a given moment — here the moment of 
true conjunction — the sine of amplitude of that point of the ecliptic 
which is then upon the eastern horizon. In the first place the longitude 
of that point (lagna) is determined, by the data and methods taught 
above, in iii. 46-48, and which are sufficiently explained in the note to 
that passage: then its sine of amplitude is found, by a process which is 
a combination of that for finding the declination from the longitude* 
and that for finding the amplitude from the declination* Thus* by ii. 28* 
R : sin gr. decl. : : sin long. : sin decl . ~, 

and, by iii, 22-23, 

sin co-lat. : R :i sin decl. s sin ampl. 
Hence, by combining terms, we have 

sin co-lat : sin gr. decl. : : sin long. : sin ampl. 

Tliis sine of amplitude receives the technical name* of udaya, or 
udayajya: the literal meaning of udaya is simply "risirg." 

4. Then, by means of the equivalents in right ascension 
(lattkodaydsavas), find the ecliptic point {lagna) called that of th& 
meridian (rnadhya) : of the declination of that point and the lati* 



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T.*.} 



Stiry&Siddhdnla. 



289 



tudeof the observer take the gam, when their direction is the 
afctfte ^otherwise;: take their difference. 

c -K .The result is the meridian zenith-distance, in degrees (natdn* 
5&)Y its flrtnc is denominated the meridian-sine (madhyajyd). . . . 

The accompanying figure (Fig. 26) will assist the comprehension of 
this and the following processes. Let N E S W he a horizontal plane, 



Fig. 26. 




N 8 the projection upon it of 
the meridian, and £ W that 
of the prime vertical, Z being 
the zenith. Let C L T be the 
ecliptic. Then C is the orient 
ecliptic-point (lagna), and C 
D the sine of its amplitude 
(udayvjyd), found by the last 
process. The meridian ecliptic 
point (madhyulagna) is L : it 
is ascertained by the method 
prescribed in iii. 49, above. 
Its distance from the zenith 
is found from its declination 
and the latitude of the place 
of observation, as taught in 
iii. 20-22; and the sine of 
that distance, by which, in 
the figure, it is seen projected, 

is Z L : it is called by the technical name madhyajyd, which we have 

translated u meridian-sine." 

5. . . . Multiply the meridian-sine by the orient-sine, and divide 
by radius: square the result, 

"' 6. And subtract it from the square of the meridian-sine: the 
8<Jitare root of the remainder is the sine of ecliptic zenith-distance 
(f'rklcshepq) ; the square root of the difference of the squares of 
that and radius is tlie sine of ecliptic-altitude (drggati). 

* Here we are taught how to find the sines of the zenith-distance and 
altitude respectively of that point of the ecliptic which has greatest alti- 
tadeyor is nearest to the zenith* and which is also the central point 
of the portion of the. ecliptic ajbove the horizon: it is called by the 
commentary, as already noticed (see note to v. 1), tribhonalayna. Thus, 
in the last figure, if.QR be the vertical circle passing through the pole 
of the ecliptic, P', and cutting the ecliptic, CT, in B, B is the central 
ecliptic-point (tribhimalagna), and the arcs seen projected in ZB and 
B R are its zenith-distance and altitude respectively. In order, now, to 
find the sine of Z B, we first find that of B L, and by the following pro- 
cess! C D is the orient-sine, already found. But since C Z and C P' 
are quadrants, C is a pole of the vertical circle Q R, and C R is a quad- 
rant. - E S is also a quadrant : take away their common part C S, and 
CE remains equal to SR, and the sine of the latter, SO, is equal to 
that of the former, C D, the " orient-sine." Now, then, Z B L is treated 



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290 E. Burgess, etc., [v. 6- 

as if it were a plane horizontal triangle, and similar to Z O S, and the 
proportion is made 

ZS:SO:ZL:BL 
or R : or.-sine : : mer.-sine : B L 

This is so far a correct process, that it gives the true sine of the arc 
BL: for, by spherical trigonometry, in the spherical triangle ZBL, 
right-angled at B, 

sin Z B L : sin B Z L : : sin arc Z L : sin arc B L 
or R:SO::ZL:sinBL 

But the third side of a plane right-angled triangle of which the sines 
of the arcs ZB and ZL are hypothenuse and perpendicular, is not the 
sine of B L. If we conceive the two former sines to be drawn from Z, 
meeting in b and I respectively the lines drawn from B and L to the 
centre, then the line joining bl will be the third side, being plainly less 
than sin B L. Hence, on subtracting sin'B L from sin*ZL, and taking 
the square root of the remainder, we obtain, not sin ZB, but a less quan- 
tity, which may readily be shown, by spherical trigonometry, to be 
sin Z B cos B L. The value, then, of the sine of ecliptic zenith-distance 
(drkkskepa) as determined by this process, is always less than the truth, 
and as the corresponding cosine (drggati) is found by subtracting the 
square of the sine from that of radius, and taking the square root of the 
remainder, its value is always proportionally greater than the truth. This 
inaccuracy is noticed by the commentator, who points out correctly its 
reason and nature : probably it was also known to those who framed the 
rule, but disregarded, as not sufficient to vitiate the general character of 
the process: and it may, indeed, well enough pass unnoticed among 
all the other inaccuracies involved in the Hindu calculations of the 
parallax. 

As regards the terms employed to express the sines of ecliptic zenith- 
distance and altitude, we have already met with the first member of each 
compound, rfr/r, literally " sight," in other connected uses : as in drgjy^ 
"sine of zenith-distance" (see above, iii. 33), drgvrtta, "vertical-circle" 
(commentary to the first verse of this chapter) : here it is combined 
with words which seem to be rather arbitrarily chosen, to form techni- 
cal appellations for quantities used only in this process: the literar 
meaning of ksJiepa'is " throwing, hurling ;" of gati, " gait, motion." 

7. The sine and cosine of meridian zenith-distance (natdneds) 
arc the approximate (asphuta) sines of ecliptic zenith-distance 
and altitude (drklcshepa, drggati). . . . 

This is intended as an allowable simplification of the above process 
for finding the sines of ecliptic zenith-distance and altitude, by substi- 
tuting for them other quantities to which they are nearly equivalent, 
and which are easier of calculation. These are the sines of zenith- 
distance and altitude of the meridian ecliptic-point (madkyalagna — L in 
Fig. 26) the former of which has already been made an element in the 
other process, under the name of "meridian-sine" (madhyajyA). It 
might, indeed, from the terms of the text, be doubtful of what point the 
altitude and zenith-distance were to be taken ; a passage cited by the 



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v. 8.] S&rya-Siddhdnta. 291 

commentator from Bhaskara's Siddhanta-Ciromani (found on page 221 
of the published edition of the Ganitadhyaya) directs the sines of zenith- 
distance and altitude of B (tribhonalagna) when upon the meridian — 
that is to say, the sine and cosine of the arc Z F — to be substituted for 
those of ZB in a hasty process: but the value of the sine would in 
this case be too small, as in the other it was too great: and as the text 
nowhere directly recognizes the point B, and as directions have been 
given in verse 5 for finding the meridian zenith-distance of L, it seems 
hardly to admit of a doubt that the latter is the point to which the text 
here intends to refer. 

Probably the permission to make this substitution is only meant to 
apply to cases where Z L is of small amount, or where C has but little 
amplitude. 

7. . . . Divide the square of the sine of one sign by the sine 
called that of ecliptic-altitude (drggatijivd) ; the quotient is the 
"divisor" (cheda). 

8. Bjr this " divisor" divide the sine of the interval between 
the meridian ecliptic-point (madhyalagna) and the sun's place : 
the quotient is to be regarded as the parallax in longitude {lam- 
bana) of the sun and moon, eastward or westward, in nfidis, etc. 

The true nature of the process by which tins final rule for finding the 
parallax in longitude is obtained is altogether hidden from sight under 
the form in which the rule is stated. Its method is as follows : 

AVo have seen, in connection with the first verse of the preceding 
chapter, that the greatest parallaxes of the sun and moon are quite 
nearly equivalent to the mean motion of each during 4 nadis. Hence, 
were both bodies in the horizon, and the ecliptic a vertical circle, the 
moon would be depressed in her orbit below the sun to an amount equal 
to her excess in motion during 4 nadis. This, then, is the moon's 
greatest horizontal parallax in longitude. To find what it would be at 
any other point in the ecliptic, still considered as a vertical circle, we 
make the proportion 

R : 4 (hor. par.) : : sin zen.-dist. : vert, parallax 

This proportion is entirely correct, and in accordance with our modern 
rule that, with a given distance, the parallax of a body varies as the sine 
of its zenith-distance : whether the Hindus had made a rigorous de- 
monstration of its truth, or whether, as in so many other cases, seeing 
that the parallax was greatest when the sine of zenith-distance was 
greatest, and nothing when this was nothing, they assumed it to vary- 
in the interval as the sine of zenith-distance, saying "if, with a sine 
of zenith-distance which is equal to radius, the parallax is four nadis, 
with a given sine of zenith-distance what is it ?" — this we will not ven- 
ture to determine. 

But now is to be considered the farther case in which the .ecliptic is 
not a vertical circle, but is depressed below the zenith a certain distance, 
measured by the sine of ecliptic zenith-distance (drktohepa), already 
found. Here again, noting that the parallax is all to be reckoned as 
parallax in longitude when the ecliptic is a vertical circle, or when the 



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292 R Burgess, etc., |V8- 

sine of ecliptic-altitude is greatest, and that it would be only parallax 
in latitude when the ecliptic should be a horizontal circle, or when the 
sine of ecliptic-altitude should be reduced to nothing, the Hindus assume 
it to vary in the interval as that sine, and accordingly make the propor- 
tion : " if, with a sine of ecliptic-altitude that is equal to radius, the par- 
allax in longitude is equal to the vertical parallax, with any given sine 
of ecliptic-altitude what is it?" — or, inverting the middle terms, 

R : sin ecl.-alt. : : vert, parallax : parallax in long. 
But we had before 

R : 4 : : sin zen.*dist. : vert, parallax 
hence, by combining terms, 

R 3 : 4 sin eel. -alt.: : sin zen.-dist. : parallax in long. 
For the third term of this proportion, now, is substituted the sine of the 
distance of the given point from the central ecliptic-point : that is to say, 
Bm (Fig. 26) is substituted for Zm; the two are in fact of equal value 
only when they coincide, or else at the horizon, when each becomes a 
quadrant; but the error involved in the substitution is greatly lessened 
by the circumstance that, as it increases in proportional amount, the 
parallax in longitude itself decreases, until at B the latter is reduced to 
nullity, as is the veitical parallax at Z. The text, indeed, as in verses 
1 and 9, puts madhynlagna, L, for tribhonalagna, B, in reckoning this 
distance : but the commentary, without ceremony or apology, reads the 
latter for the former. These substitutions being made, and the propor- 
tion being reduced to the form of an equation, we have 

. , sin dist.X 4 pin ecl.-alt. 
par. in long, as — 

which reduces to 

sin dist. Aa . sin dint, 

or 



K f -r- 4 sin eel. alt. JR*-r sin ecl.-alt. 

and since ±R*=(£R) 2 , and £R = sin30°, we have finally 

. . sin dirt. 

par. in ng. — — — ^ ^ ^ ^ 

which is the rule given in the text To the denominator of the fraction, 
in its final form, is given the technical name of cheda, " divisor/ 9 which 
word we have had before similarly used, to designate one of the factors 
in a complicated operation (see above, iii. 35, 38). 

We wul now examine the correctness of the second principal propor- 
tion from which the rule is deduced. It is, in terms of the last figure 
(Fig. 26), 

R : sin ZP' (=BR) : : m M : m n 

Assuming the equality of the little triangles M m n and M m n', and 
accordingly that of the angles m M n and M m n', which latter equals 
Z m P', we have, by spherical trigonometry, as a true proportion, 

sin m n 1 M : sin M m n' : : m M : m n' 
or R:sin ZwF: :mM:mn 

Hence the former proportion is correct only when sin ZP'and sin 
ZwF are equal ; that is to say, when Z P / measures the angle ZmF; 



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r. : 10.] S&rya-Siddhdnta. . 293 

and this can be the case only when Z m, as well as P' m y is a quadrant, 
or when m is on the horizon. Here again, however, precisely as in the 
case last noticed, the importance of the error is kept within very narrow 
limits by the fact that, as its relative consequence increases, the amount 
of the parallax in longitude affected by it diminishes. 

: 9. When the sun's longitude is greater than that of the meri- 
dian ecliptic-point (madhyalagna), subtract the parallax in longi- 
tude from the end of the lunar day ; -when less, add the same : 
repeat the process until all is fixed. 

The text so pertinaciously reads "meridian ecliptic-point" (madhya- 
lagna) where we should expect, and ought to have, " central ecliptic- 
point " (tribhonalagna\ that we are almost ready to suspect it of mean- 
ing to designate the latter point by the fonner name. It is sufficiently 
clear that, whenever the sun and moon are to the eastward of the cen- 
tral ecliptic-point, the effect of the parallax in longitude will be to throw 
the moon forward on her orbit beyond the sun, and so to cause the time 
of apparent to precede that of real conjunction; and the contrary. 
Hence, in the eastern hemisphere, the parallax, in time, is subtractive, 
while in the western it is additive. But a single calculation and appli- 
cation of the correction for parallax is not enough ; the moment of ap- 
parent conjunction must be found by a series of successive approxima- 
tions : since if, for instance, the moment of true conjunction is 25 n 2 T , 
and the calculated parallax in longitude for that moment is 2 n 21 r , the 
apparent' end of the lunar day will not be at 27 n 23 v , because at the 
latter time the parallax will be greater than 2 n 21 T , deferring accordingly 
still farther the time of conjunctions-anil so on. The commentary ex- 
plains the method of procedure more fully, as follows: for the moment 
of true conjunction in longitude calculate the parallax in longitude,- and 
apply it to that moment: for the time thus-found calculate the parallax 
anew, and apply it to the moment of true conjunction : again, for the 
time found a* the result of this process, calculate the parallax, and ap>» 
ply it as before ; and so proceed, until a moment is arrived at, at which 
the difference in actual longitude, according to the motions of the two 
planets, will just equal and counterbalance the parallax in longitude. 
; The accuracy of this approximative process cannot but be somewhat 
Jmpaired by the circumstance that, while the parallax is reckoned in 
difference of mean motions, the corrections of longitude must be made 
in .true motions. Indeed, the reckoning of the horizontal parallax in 
time as 4 nadis, whatever be the rate of motion of the sun and moon, is 
one of the most palpable among the many errors which the Hindu pro- 
cess involves. 

, To ascertain the moment of apparent conjunction in longitude, only 
the parallax in longitude requires to be known ; but to determine the 
xitne of occurrence of the other phases of the eclipse, it is necessary to 
take into account the parallax in latitude, the ascertainment of which is 
accordingly made the subject of the next rale. 

,.10. If the sine of ecliptic zenith-distance (drkkshepa) be multi- 
plied by the difference of the mean motions of the sun and 
vol. ti. 38 



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294 E. Burgees, etc., [t. 10- 

moon, and divided by fifteen times radius, the result will be]the 
parallax in latitude (avanati). 

As the sun's greatest parallax is equal to the fifteenth part of his 
mean daily motion, and that of the moon to the fifteenth part of hers 
(see note to iv. 1, above), the excess of the moon's parallax over that of 
the sun is equal, when greatest, to one fifteenth of the difference of 
their respective mean dally motions. This will be the value of th* 
parallax in latitude when the ecliptic coincides with the horizon, or 
when the sine of ecliptic zenith-distance becomes equal to radius. On 
the other hand, the parallax in latitude disappears when this same sine 
is reduced to nullity. Hence it is to be regarded as varying with the 
sine of ecliptic zenith-distance, and, in order to find its value at any 
given point, we say i4 if, with a sine of ecliptic zenith-distance which is 
equal to radius, the parallax in latitude is ono fifteenth of the difference 
of mean daily motions, with a given sine of ecliptic zenith-distance 
what is it ? w or 

R : cliff, of mean m.-~15 : : sin eel. zen.-dist : parallax in lat 

This proportion, it is evident, would give with entire correctness the 
parallax at the central ecliptic-point (B in Fig. 26), where the whole 
vertical parallax is to be reckoned as parallax in latitude. But the rule 
given in the text also assumes that, with a given position of the ecliptic, 
the parallax in latitude is the same at any point in the ecliptic. Of this 
the commentary offers no demonstration, but it is essentially true. For, 
regarding the little triangle Mmnasa plane triangle, right-angled at*» 
and with its angle n w M equal to the angle Z m B, we have 

Rrsin ZmB::Mm:M« 
But, in the spherical triangle Z m B, right-angled at B, 

R : sin Z m B : : sin Z m : sin Z B 
Hence, by equality of ratios, 

sin Z m : sin Z B : ; M m : M ft 
But, as before shown, 

R r sin Zm : : gr. parallax : M m 
Hence, by combining terms, 

R : sin Z B : : gr. parallax : M n 

Tli at is to say, whatever be the position of m, the point for which (ha 

parallax in latitude is sought, this will be equal to the product of the 

greatest parallax into the sine of ecliptic zenith-distance, divided by 

radius : or, as the greatest parallax equals the difference of mean mo* 

tions divided by fifteen, 

. , . siaecl.zen.-di8t.Xdiftofm.in.-rL5 •inecl.sen.-dutt.Xdiff of m.m. 
par. fa, at = R or ^ 

The next verse teaches more summary methods of arriving at the 
same quantity. 

11. Or, the parallax in latitude ia the quotient arising from 
dividing the sine of ecliptic zenith-distance {drldahepa) by sev- 



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v. 13.] S&rya-Siddhdnta. 295 

enty, or, from multiplying it by forty-nine, and dividing it by 
radius. 

In the expression given above for the value of the parallax in latitude, 
all the terms are constant excepting the sine of ecliptic zenith-distance. 
The difference of the mean daily motions is 731' 27", and fifteen times 
radius is 51,570'. Now 731' 27"-r51,570' equals Ta ^ or 48.77—R; 
to which the expressions given in the text are sufficiently near approxi- 
mations. 

12. The parallax in latitude is to be regarded as south or 
north according to the direction of the meridian-sine (madkyajyd). 
When it and the moon's latitude are of like direction, take their 
sum ; otherwise, their difference : 

13. With this calculate the half-duration (s&iti), half total ob- 
scuration (vimarda), amount of obscuration {gr&sa), etc., in the 
manner already taught; likewise the scale of projection (pra* 
mdna), the deflection (valana), the required amount of obscura- 
tion, etc., as in the case of a lunar eclipse. 

In ascertaining the true time of occurrence of the various phases of 
a solar eclipse, as determined by the parallax of the given point of ob- 
servation, we are taught first to make the whole correction for parallax 
in latitude, and then afterward to apply that for parallax in longitude. 
The former part of the process is succinctly taught in verses 12 and 13 : 
the rules for the other follow in the next passage. The language of the 
text, as usual, is by no means so clear and explicit as could be wished. 
Thus, in the case before us, we are not taught whether, as the first step 
in this process of correction, we are to calculate the moon's parallax in 
latitude for the time of true conjunction (tithyanta, " end of the lunar 
day "), or for that of apparent conjunction (madhyagrahancL, u middle of 
the eclipse "). It might be supposed that, as we have thus far only had 
in the text directions for finding the sine and cosine of ecliptic zenith- 
distance at the moment of true conjunction, the former of them was to 
be used in the calculations of verses 10 and 11, and the result from it, 
which would be the parallax at the moment of true conjunction, applied 
here as the correction needed. Nor, so far as we have been able to 
discover, does the commentator expound what is the true meaning of 
the text upon this point It is sufficiently evident, however, that the 
moment of apparent conjunction is the time required. We have found, 
by a process of successive approximation, at what time (see Fig. 25), the 
moon (her latitude being neglected) being at m and the sun at n, the 
parallax in longitude and the difference of true longitude will both be 
the same quantity, tnn, and so, when apparent conjunction will take 
place. Now, to know the distance of the two centres at that moment, 
we require to ascertain the parallax in latitude, n M, for the moon at m, 
and to apply it to the moon's latitude when in the same position, taking 
their sum when their direction is the same, and their difference when 
their direction is different, as prescribed by the text ; the net result will 
be the distance required. The commentary, it may be remarked, ex* 
pressly states that the moon's latitude is to be calculated in this opera- 



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296 E. Burgess, etc., [v. 13-* 

tion for the time of apparent conjunction (mndhyagrakana). The dis- 
tance thus found will determine the amount of greatest obscuration, and 
the character of the eclipse, as taught in verso 10 of the preceding chap- 
ter. It is then farther to be taken as the foundation of precisely such a 
process as that described in verses 12-15 of the same chapter, in order 
to ascertain the half-time of duration, or of total obscuration ; that is to 
say, the distance in latitude of the two centres being first assumed as 
invariable through the whole duration of the eclipse, the half-time of 
duration, and the resulting moments of contact and separation are to be 
ascertained : for these moments the latitude and parallax in latitude are 
to be calculated anew, and by them a new determination of the times of 
contact and separation is to be made, and so on, until these are fixed 
with the degree of accuracy required. If the eclipse be total, a similar 
operation must be gone through with to ascertain the moments of im- 
mersion and emergence. No account is made, it will be noticed, of the 
possible occurrence of an anuular eclipse. 

The intervals thus found, after correction for parallax in latitude 
only, between the middle of the eclipse and the moments of contact and 
separation respectively, are those which are called in the last chapter 
(vv. 19, 23), the "mean half-duration" (madkyasihityardha). 

In this process for finding the net result, as apparent latitude, of the 
actual latitude and the parallax in latitude, is brought out with dis- 
tinctness the inaccuracy already alluded to; that, whatever be the 
moon's actual latitude, her parallax is always calculated as if she were 
in the ecliptic. In an eclipse, however, to which case alone the Hindu 
processes are intended to be applied, the moon's latitude can never be 
of any considerable amount 

The propriety of determining the direction of the parallax in latitude 
by means of that of the meridian-sine (ZL in Fig. 26), of which the 
direction is established as south or north by the process of its calcula- 
tion, is too evident to call for remark. 

In verse 13 is given a somewhat confused specification of matters 
which are, indeed, affected by the parallax in latitude, but in different 
modes and degrees. The amount of greatest obscuration, and the 
(mean) half-times of duration and total obscuration, are the quantities 
directly dependent upon the calculation of that parallax, aa here pre- 
sented : to find the amount of obscuration at a given moment— as also 
the time corresponding to a given amount of obscuration— -we require 
to know also the true half-duration, as found by the rules stated in the 
following passage: while the scale of projection and the deflection. are 
affected by parallax only so far as this alters the time of occurrence of 
the phases of the eclipse, 

14. For the end of the lunar day, diminished and increased by 
the half-duration, as formerly, calculate again the parallax, in 
longitude for the times of contact {grdsci) and of separation (mok- 
sha), and find the difference between these and tue parallax in 
longitude (harija) for the middle of the eclipse. 

15. If, in the eastern hemisphere, the parallax in. longitude 
for the contact is greater than that for the middle, and that ibr 



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t. 1 ».] Surya-Siddli&nta. 297 

the separation less ; and if, in the western hemisphere, the con- 
trary is the case — 

16. Then the difference of parallax in longitude is to be added 
to the half-duration on the side of separation, and likewise on 
that of contact (pragrahaya) ; when the contrary is true, it is to 
be subtracted. 

17. These rules are given for cases where the two parallaxes 
are in the same hemisphere : where they are in different hemi- 
spheres, the sum of the parallaxes in longitude is to be added to 
tne corresponding half-auration. The principles here stated ap- 
ply also to the half-time of total obscuration. 

We are supposed to have ascertained, by the preceding process, the 
true amount of apparent latitude at the moments of first and last con- 
tact of the eclipsed and eclipsing bodies, and consequently to hare de- 
termined the dimensions of the triangle — corresponding, in a solar 
eclipse, to CGP, Fig. 21, in a lnnar — made np of the latitude, the dis- 
tance in longitude, and the sum of the two radii. The question now is 
how the duration of the eclipse will be affected by the parallax in longi- 
tude. If this parallax remained constant during the continuance of the 
eclipse, its effect would be nothing ; and, having once determined by it 
the time of apparent conjunction, we should not need to take it farther 
into account But it varies from moment to moment, and the effect of 
its variation is to prolong the duration of every part of a visible eclipse. 
For, to the east of the central ecliptic-point, it throws the moon's disk 
forward upon that of the sun, thus hastening the occurrence of all the 
phases of the eclipse, but by an amount which is all the time decreasing, 
so that it hastens the beginning of the eclipse more than the middle, 
and the middle more than the close : to the west of that same point, on 
the other hand, it depresses the moon's disk away from the sun's, but by 
an amount constantly increasing, so that it retards the end of the eclipse 
more than its middle, and its middle more than its beginning. The 
effect of the parallax in longitude, then, upon each half-duration of the 
eclipse, will be measured by the difference between its retarding and ac- 
celerating effects upon contact and conjunction, and upon conjunction 
and separation, respectively : and the amount of this difference will 
always be additive to the time of half-duration as otherwise determined. 
If, however, contact and conjunction, or conjunction and separation, 
take place upon opposite sides of the point of no parallax in longitude, 
then the sum of the two parallactic effects, instead of their difference, 
will be to be added to the corresponding half-duration : since the one, 
on the east, will hasten the occurrence of the former phase, while the 
other, on the west, will defer the occurrence of the latter phase. The 
amount of the parallax in longitude for the middle of the eclipse has 
already been found ; if, now, we farther determine its amount — reckoned, 
it will be remembered, always in time — for the moments of contact and 
separation, and add the difference or the sum of each of these and the 

Sarallax for the moment of conjunction to the corresponding half- 
oration as previously determined, we shall have the true times of half- 
duration. In order to find the parallax for contact and separation, we 



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208 JR Burgess, etc., [v. 17- 

repeat the same process (see above, v. 9) by which that for conjunction 
was found : as we then started from the moment of true conjunction,' 
and, by a series of successive approximations, ascertained the time when 
the difference of longitude would equal the parallax in longitude, so now 
we start from two moments removed from that of true conjunction by 
the equivalents in time of the two distances in longitude obtained by the 
last process, and, by a similar series of successive approximations, ascer- 
tain the times when the differences of longitude, together with the par- 
allax, will equal those distances in longitude. 

In the process, as thus conducted, there is an evident inaccuracy. Tt 
is not enough to apply the whole correction for parallax in latitude, and 
then that for parallax in longitude, since, by reason of the change effected 
by the latter in the times of contact and separation, a new calculation of 
the former becomes necessary, and then again a new calculation of the 
latter, and so on, until, by a series of doubly compounded approxima- 
tions, the true value of each is determined. This was doubtless known 
to the framers of the system, but passed over by them, on account of 
the excessively laborious character of the complete calculation, and be- 
cause the accuracy of such results as they could obtain was not sensibly 
affected by its neglect 

The question naturally arises, why the specifications of verse 15 are 
made hypothetical instead of positive, and why, in the latter half of 
verse 16, a case is supposed which never arises. The commentator an- 
ticipates this objection, and takes much pains to remove it : it is not 
worth while to follow his different pleas, which amount to no real expla- 
nation, saving to notice his last suggestion, that, in case an eclipse begins . 
before sunrise, the parallax for its earlier phase or phases, as calculated 
according to the distance in time from the lower meridian, may be less, 
than for its later phases — and the contrary, when the eclipse' ends after 
sunset. This may possibly be the true explanation, although we are 
justly surprised at finding a case of so little practical consequence, and 
to which no allusion has been made in the previous processes, here 
taken into account. 

The text, it may be remarked, by its use of the terms " eastern and 
western hemispheres" (kapdla, literally "cup, vessel"), repeats once more 
its substitution of the meridian ecliptic-point (madkyalagna) for the 
central ecliptic-point (tribhonalagna), as that of no parallax in longitude; . 
the meridian forming the only proper and recognized division, of tbe 
heavens into an eastern and a western hemisphere. 

We are now prepared to see the reason of the special directions given 
in verses 19 and 23 of the last chapter, respecting the reduction, in a 
solar eclipse, of distance in time from the middle of the eclipse to dis- 
tance in longitude of the two centres. The "mean half-duration." 
(madhyasthityardha) of the eclipse is the time during which the true dis- 
tance of the centres at the moments of contact or separation, as found 
by the process prescribed in verses 12 and 13 of this chapter, would be 
gained by the moon with her actual excess of motion, leaving out of ac- 
count the variation of parallax in longitude : the " true half-duration " 
(ftpkutaslkityardha) is the increased time in which, owing to that varia- 
tion, the same distance in longitude is actually gained by the moon ; 



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vi. 1.] S&rya-Siddhdnta. 299 

the effect of the parallax being equivalent either to a diminution of the 
moon's excess of motion, or to a protraction of the distance of the two 
centers — both of them in the ratio of the true to the mean half-duration. 
If then, for instance, it he required to know what will be the amount of 
obscuration of the sun half an hour after the first contaet, we shall first 
subtract this interval from the true hsJf»duration before conjunction ; the 
remainder will be the actual interval to the middle of the eclipse : this 
interval, then, we shall reduce to its value as distance in longitude by 
diminishing it, either before. or after its reduction to minutes of are, in 
the ratio of the true to the mean halfduration. The rest of the process 
will be oerformed precisely as in the case of an eclipse of the moon. 

Notwithstanding the. ingenuity and approximate correctness of many 
of the rules and methods of calculation taught in this chapter, the whole 
process for the ascertainment of parallax contains so many elements of 
error that it hardly deserves to be called otherwise than cumbrous and 
bungling. The false estimate of the difference between the sun's and 
moon's horizontal parallax — the neglect, in determining it, of the varia* 
tion of the moon's distance — the estimation of its value in time made 
always according to mean motions, whatever be the true motions of the 
planets at the moment — the neglect, in calculating the amount of par- 
allax, of the moon's latitude— these, with all the other inaeeuracies of 
the processes of calculation which have been pointed out in the notes, 
render it impossible that the results obtained should ever be more than 
a rude approximation to the truth. 

In farther illustration of the subject of solar eclipses, as exposed in 
this and the preceding chapters, we present, in the Appendix, a full cal- 
culation of the eclipse of May 26th, 1854, mainly as made for the trans- 
lator, during his residence in India, by a native astronomer. 

CHAPTER VI. i Juihliiry. 

OP THE PBOJECTION OF ECLIPSES. ^*sOf Csllfbfnl^ 

Otumu M , value of a projection; 2-4, general directions; 5-6, how tolayofl 
the deflection sad latitude for the beginning and end of the eclipse ; T t to exhibit 
the points of contact and separation ; 8-10, how to lay off the deflection and lati- 
tude for the middle of the eclipse ; 11, to show the amount of greatest obscura- 
tion; 12, reversal of directions in the western hemisphere ; 13, least amount of 
obscuration observable; 14-16, to draw the path of the eclipsing bod/; 17-19, to 
show the amount of obscuration at a given time ; 20-22, to exhibit the points of 
immersion and emergence in a total eclipse ; 23, color of the part of the moon 
obscured ; 24, caution as to communicating a knowledge of these matters. 

1. Since, without a projection (chedyaka), the precise (sphutai 
differences of the two eclipses are not understood, I shall prexseea 
to explain the exalted doctrine of the projection. 



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800 E. Burgess, etc., [vi. 1- 

The term chedyaka is from the root chid, " split, divide, sunder," and 
indicates, as here applied, the instrumentality by which distinctive dif- 
ferences are rendered evident The name of the chapter, parilekhddki- 
k&ra, is not taken from this word, but from parilekha, " delineation, 
figure," which occurs once below, in the eighth verse. 

2. Having fixed, upon a well prepared surface, a point, de- 
scribe from it, in the first place, with a radius of forty-nine digits 
(angula), a circle for the deflection (valana) : 

3. Then a second circle, with a radius equal to half the sum of 
the eclipsed and eclipsing bodies ; this is called the aggregate- 
circle (saindsa) ; then a third, with a radius equal to half the 
eclipsed body. 

4. The determination of the directions, north, south, east, and 
west, is as formerly. In a lunar eclipse, contact (grahana) takes 

1)lace on the east, and separation (moksha) on the west ; in a so- 
ar eclipse, the contrary. 

The larger circle, drawn with a radius of about three feet, is used solely 
in laying off the deflection (valana) of the ecliptic from an east and 
west circle. We have seen above (iv. 24, 25) that the sine of this de- 
flection was reduced to its value in a circle of forty-nine digits' radius, 
by dividing by seventy its value in minutes. The second circle is em- 
ployed (see below, vv. 6, 7) in determining the points of contact and 
separation. The third represents the eclipsed body itself, always main- 
taining a fixed position in the centre of the figure, even though, in a 
lunar eclipse, it is the body which itself moves, relatively to the eclipsing 
shadow. For the scale by which the measures of the eclipsed and 
eclipsing bodies, the latitudes, etc., are determined, see above, iv. 26. 

The method of laying down the cardinal directions is the same with 
that used in constructing a dial ; it is described in the first passage of 
the third chapter (iii. 1-4). 

The specifications of the latter half of verse 4 apply to the eclipsed 
body, designating upon which side of it obscuration will commence and 
terminate. 

5. In a lunar eclipse, the deflection (valana) for the contact is 
to be laid off in its own proper direction, but that for the separa- 
tion in reverse ; in an eclipse of the sun, the contrary is the case. 

The accompanying figure (Fig. 27) will illustrate the Hindu method 
of exhibiting, by a projection, the various phases of an eclipse. Its 
conditions are those of the lunar eclipse of Feb. 6th, 1860, as deter- 
mined by the data and methods of this treatise : for the calculation see 
the Appendix. Let M be the centre of the figure and the place of the 
moon, and let N S and E W be the circles of direction drawn through 
the moon's centre ; the former representing (see above, under iv. 24, 25) 
a great circle drawn through the north and south points of the horizon, 
the latter a small circle parallel to the prime vertical. In explanation 
of the manner in which these directions aie presented by the figure, we 
would remark that we have adapted it to a supposed position of the 



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IS.] 



Surya-Siddlidiita, 



801 



observer on the north side of his projection, as at N, and looking south- 
ward — a position which, in our latitude, he would naturally assume, tor 



Fig. 27. 




the purpose of comparing the actual phases of the eclipse, as they oc- 
curred, with his delineation of them. The heavier circle, I /', is that 
drawn with the sum of the semi-diameters, or the u aggregate-circle ;" 
while the outer one, N E S W, is that for the deflection. This, in order 
to reduce the size of the whole figure, we have drawn upon a scale very 
much smaller than that prescribed ; its relative dimensions being a mat- 
ter of no consequence whatever, provided the sine of the deflection be 
made commensurate with its radius. In our own, or the Greek, method 
of laying off an arc, by its angular value, the radius of the circle of de- 
flection would also be a matter of indifference : the Hindus, ignoring 
angular measurements, adopt the more awkward and bunding method 
of laying off the arc by means of its sine. Let v w equal the deflection, 
calculated for the moment of contact, expressed as a sine, and in terms 
of a circle in which £ M is radius. Now, as the moon's contact with 
the shadow takes place upon her eastern limb, the deflection for the 
contact must be laid off from the east point of the circle ; and, as the 
calculated direction of the deflection indicates in what way the ecliptic 
is pointing eastwardly, it must be laid off from E in its own proper di- 
vou vi. 39 



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802 R Burgess, etc., [vi. 5- 

rection. In the case illustrated, the deflection for the contact is north : 
hence we lay it off northward from E, and then the line drawn from M 
to v, its extremity — which line represents the direction of the ecliptic 
at the moment — points northward. Again, upon the side of separation 
— which, for the moon, is the western side — we lay off the deflection for 
the moment of separation : but we lay it off from W in the reverse of 
its true direction, in order that the line from its extremity to the centre 
may truly represent the direction of the ecliptic. Thus, in the eclipse 
figured, the deflection for separation is south ; we lay it off northward 
from W, and then the line v* M points, toward M, southward. In a solar 
eclipse, in which, since the sun s western limb is the first eclipsed, the 
deflection for contact must be laid off from W, and that for separation 
from E, the direction of the former requires to be reversed, and that of 
the latter to be maintained as calculated. 

6. From the extremity of either deflection draw a line to the 
centre: from the point where that cuts the aggregate-circle 
(samdsa) are to be laid off the latitudes of contact and of separa- 
tion. 

7. From the extremity of the latitude, again, draw a line to 
the central point : where that, in either case, touches the eclipsed 
body, there point out the contact and separation. 

8. Always, in a solar eclipse, the latitudes are to be drawn in 
the figure (parileklia) in their proper direction; in a lunar 
eclipse, in the opposite direction. . . . 

The lines v M and v* M, drawn from v and i/, the extremities of the 
nines or arcs which measure the deflection, to the centre of the figure, 
represent, as already noticed, the direction of the ecliptic with reference 
to an east and west line at the moments of contact and separation. 
From them, accordingly, and at right angles to them, are to be laid off 
the values of the moon's latitude at those moments. Owing, however, 
to the principle adopted in the projection, of regarding the eclipsed 
body as fixed in the centre of the figure, and the eclipsing body as pass- 
ing over it, the lines v M and v' M do not, in the case of a lunar eclipse, 
represent the ecliptic itself in which is the centre of the shadow, hut the 
small circle of latitude, in which is the moon's centre : hence, in laying 
off the moon's latitude to determine the centre of the shadow, we re- 
verse its direction. Thus, in the case illustrated, the moon's latitude is 
always south : we lay oft, then, the lines k I and k' l\ representing its 
value at the moments of contact and separation, northward : they are, 
like the deflection, drawn as sines, and in such manner that their ex- 
tremities, I and l' y are in the aggregate-circle : then, since I M and V M 
are each equal to the sum of the two semi-diameters, and Ik and V k f 
to the latitudes, k M and k r M will represent the distances of the centres 
in longitude, and I and /' the places of the centre of the shadow, at con- 
tact and separation : and upon describing circles from I and /', with radii 
equal to the semi-diameter of the shadow, the points c and *, where 
these touch the disk of the moon, will be the points of first and last con- 
tact : e and s being also, as stated in the text, the points where I M and 
/' M meet the circumference of the disk of the eclipsed body. 



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vi. 11.] S&rya-Siddhdnta. 808 

8. . . . In accordance with this, then, for the middle of the 
eclipse, 

9. The deflection is to be laid off— eastward, when it and the 
latitude are of the same direction; when they are of different 
directions, it is to be laid off westward: this is for a lunar 
eclipse ; in a solar, the contrary is the case. 

10. From the end of the deflection, again, draw a line to the 
central point, and upon this line of the middle lay off the lati- 
tude, in the direction of the deflection. 

11. From the extremity of the latitude describe a circle with 
a radius equal to half the measure of the eclipsing body : what- 
ever of the disk of the eclipsed body is enclosed within that 
circle, so much is swallowed up by the darkness (tamos). 

The phraseology of the text in this passage is somewhat intricate and 
obscure ; it is fully explained by the commentary, as, indeed, its mean- 
/ ing is also deducible with sufficient clearness from the conditions of the 
problem sought to be solved. It is required to represent the deflection 
of the ecliptic from an east and west line at the moment of greatest 
obscuration, and to fix the position of the centre of the eclipsing body 
at that moment The deflection is this time to be determined by a 
secondary to the ecliptic, drawn from near the north or aputh point of 
the figure. The first question is, from which of these two points shall 
the deflection be laid off, and the line to the centre drawn. Now since, 
according to verse 10, the latitude itself is to be measured upon the line 
of deflection, the latter must be drawn southward or northward accord- 
ing to the direction in which the latitude is to be laid off. And this is 
the meaning of the last part of verse 8; "in accordance," namely, with 
the direction in which, according to the previous part of the verse, the 
latitude is to be drawn. But again, in which direction from the north 
or south point, as thus determined, shall the deflection be measured? 
This must, of course, be determined by the direction of the deflection 
itself: if south, it must obviously be measured east from the north point 
and west from the south point ; if north, the contrary. The rules of 
the text are in accordance with this, although the determining circum- 
stance is made to be the agreement or non-agreement, in respect to 
direction, of the deflection with the moon's latitude— the latter being 
this time reckoned in its own proper direction, and not, in a lunar 
eclipse, reversed. Thus, in the case for which the figure is drawn, as 
the moon's latitude is south, and must be laid off northward from M, 
the deflection, v H 10", is measured from the north point ; as deflection 
and latitude are both south, it is measured east from N. In an eclipse 
of the sun, on the other hand, the moon's latitude would, if north, be 
laid off northward, as in the figure, and hence also, the deflection would 
be measured from the north point : but it would be measured eastward, 
if its own direction were south, or disagreed with that of the latitude. 

The line of deflection, which is M v" in the figure, being drawn, and 
having the direction of a perpendicular to the ecliptic at the moment of 
opposition, the moon's latitude for that moment, M i", is laid off directly 



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804 K Burgess, ttc. % [vi. 11- 

upon it The point I" is, accordingly, the position of the centre of the 
shadow at the middle of the eclipse, and if from that centre, with a 
radius equal to the semi-diameter of the eclipsing body, a circle be drawn, 
it will include so much of the disk of the eclipsed body as is covered 
when the obscuration is greatest In the figure the eclipse is shown as 
total, the Hindu calculations making it so, although, in fact, it is only a 
partial eclipse. 

12. By the wise man who draws the projection (chedyahz), 
upon the ground or upon a board, a reversal of directions is to 
be made in the eastern and western hemispheres. 

This verse is inserted here in order to remove the objection that, in 
the eastern hemisphere, indeed, all takes place as stated, but, if the 
eclipse occurs west of the meridian, the stated directions require to be 
all of them reversed. In order to understand this objection, we must 
take notice of the origin and literal meaning of the Sanskrit words 
which designate the cardinal directions. The face of the observer is 
supposed always to be eastward : then " east" is pr&Hc, a forward, toward 
the front"; u west" is pafe&t, "backward, toward the rear" : "south" is 
dakshina, " on the right" ; " north" is uttara, " upward" (i. e., probably, 
toward the mountains, or up the course of the rivers in north-western 
India). These words apply, then, in etymological strictness, only when 
one is looking eastward — and so, in the present case, only when the 
eclipse is taking place in the eastern hemisphere, and the projector is 
watching it from the west side of his projection, with the latter before 
him : if, on the other hand, he removes to E, turning his face westward, 
and comparing the phenomena as they occur in the western hemisphere 
with his delineation of them, then " forward" (pr&ftc) is no longer east, 
but west; "right" (dakshina) is no longer south, but north, etc. 

It is unnecessary to point out that this objection is one of the most 
frivolous and hair-splitting character, and its removal by the text a waste 
of trouble : the terms in question have fully acquired in the language an 
absolute meaning, as indicating directions in space, without regard to the 
position of the observer. 

13. Owing to her clearness, even the twelfth part of the moon, 
when eclipsed (grasta), is observable ; but, owing to his piercing 
brilliancy, even three minutes of the sun, when eclipsed, are not 
observable. 

The commentator regards the negative which is expressed in the lat- 
ter half of this verse as also implied in the former, the meaning being 
that an obscuration of the moon s disk extending over only the twelfth 
part of it does not make itself apparent We have preferred the inter- 
pretation given above, as being better accordant both with the plain and 
simple construction of the text and with fact 

14. At the extremities of the latitudes make three points, of 
corresponding names ; then, between that, of the contact and 



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tL 22.] SCLrya-Siddlidnta. 805 

that of the middle, and likewise between that of the separation 
and that of the middle, 

15. Describe two fish-figures (mctUya): from the middle of 
these having drawn out two lines projecting through the mouth 
and tail, wherever their intersection takes place, 

16. There, with a line touching the three points, describe an 
arc : that is called the path of the eclipsing body, upon ' which 
the latter will move forward. 

The deflection and the latitude of three points in the continuance of 
the eclipse having been determined and laid down upon the projection, 
it is deemed unnecessary to take the same trouble with regard to any 
other points, these three being sufficient to determine the path of the 
eclipsing body : accordingly, an arc of a circle is drawn through them, 
and is regarded as representing that path. The method of describing 
the arc is the same with that which has already been more than once 
employed (see above, iii. 1-4, 41-42) : it is explained here with some- 
what more fullness than before. Thus, in the figure, J, /", and /' are the 
three extremities of the moon's latitude, at the moments of contact, 
opposition, and separation, respectively : we join I /", I" /', and upon 
these lines describe fish-figures (see note to iii. 1-5) ; their two extremi- 
ties ("mouth 1 ' and "tail") are indicated by the intersecting dotted lines 
in the figure: then, at the point, not included in the figure, where the 
lines drawn through them meet one another, is the centre of a circle 
passing through 2, I", and V. 

17. From half the sum of the eclipsed and eclipsing bodies 
subtract the amount of obscuration, as calculated for any given 
time : take a little stick equal to the remainder, in digits, and, 
from the central point, 

18- Lay it off toward the path upon either side — when the 
time is before that of greatest obscuration, toward the side of 
contact; when the obscuration is decreasing, in the direction of 
separation — and where the stick and the path of the eclipsing 
body 

19. Meet one another, from that point describe a circle with a 
radius ec[ual to half the eclipsing bcxly : whatever of the eclipsed 
body is included within it, that point out as swallowed up by 
the darkness (tamas). 

20. Take a little stick equal to half the difference of the 
measures (mdna) f and lay it off in the direction of contact, calling 
it the stick of immersion (nimilana) : where it touches the path, 

21. From that point, with a radius equal to half the eclipsing 
body, draw a circle, as in the former case : where this meets the 
circle of the eclipsed body, there immersion takes place. 

22. So also for the emergence (unmilana), lay it off in the 
direction of separation, and describe a circle, ad before : it will 
show the point of emergence in the manner explained. . 



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306 E. Burgess, etc., [vi. 22- 

The method of these processes is so cleaf as to call for no detailed 
explanation. The centre of the eclipsing body being supposed to be 
alwajB in the arc 1 1" l\ drawn as directed in the last passage, we have 
only to fix a point in this arc which shall be at a distance from M cor- 
responding to the calculated distance of the centres at the given time, 
and from that point to describe a circle of the dimensions of the eclipsed 
body, and the result will be a representation of the then phase of the 
eclipse. If the point thus fixed be distant from M by the difference of 
the two semi-diameters, as M t 7 , M «*, the circles described will touch the 
disk of the eclipsed body at the points of immersion and emergence, 
t and e. 

23. The part obscured, when less than half, will be dusky 
(sadh&mra) ; when more than half, it will be black ; when emerg- 
ing, it is dark copper-color (krshnatdmra) ; when the obscuration 
is total, it is tawny (kapila). 

The commentary adds the important circumstance, omitted in the 
text, that the moon alone is here spoken of; no specification being 
added with reference to the sun, because, in a solar eclipse, the part 
obscured is always black. 

A more suitable place might have been found for this verse in the 
fourth chapter, as it has nothing to do with the projection of an eclipse. 

24. This mystery of the gods is not to be imparted indiscrim- 
inately : it is to be made known to the well-tried pupil, who 
remains a year under instruction. 

The commentary understands by this mystery, which is to be kept 
with so jealous care, the knowledge of the subject of this chapter, the 
delineation of an eclipse, and not the general subject of eclipses, as 
treated in the past three chapters. It seems a little curious to find a 
matter of so subordinate consequence heralded so pompously in the 
first verso of the chapter, and guarded so cautiously at its close. 



CHAPTER VII. 
OF PLANETABY CONJUNCTIONS. 

Vmmum s — 1, general classification of planetary conjunctions; 2-6, method of de- 
termining at what point on the ecliptic, and at what time, two planets will come 
to have the tame longitude ; 7-10, how to find the point on the ecliptic to which 
a pi limit, having latitude, will be referred by a circle passing through the north 
nod south points of the horizon ; 11, when a planet must be so referred ; 12, how 
to WKwrtain the interval between two planets when in conjunction upon such a 
north ami south line; 18-14, dimensions of the lesser planets; 15-18, modes of 
#s MM ting the coincidence between the calculated and actual places of the planets ; 
10 20, definition of different kinds of conjunction; 20-21, when a planet, in con- 



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vii. 6.] S&rya-Siddhdnta. 307 

junction, is vanquished or victor; 22, farther definition of different kinds of con- 
junction; 23, usual prevalence of Venus in a conjunction; 23, planetary conjunc- 
tions with the moon; 24, conjunctions apparent only; -why calculated. 

1. Of the star-planets there take place, with one another, 
encounter (yvddha) and conjunction (samdgama) ; with the moon, 
conjunction (samdgama) ; with the sun, heliacal setting (astamana). 

The "star-planets" (tdrdgraha) are, of course, the five lesser planets, 
exclusive of the sun and moon. Their conjunctions with one another 
and with the moon, with the asterisms (nalcshatra), and with the sun, 
are the subjects of this and the two following chapters. 

For the general idea of u conjunction" various terms are indifferently 
employed in this chapter, as samdgama, " coming together", samyoga, 
u conjunction," yoga, "junction" (in viii. 14, also, melaka, " meeting") : 
the word yuti, " union," which is constantly used in the same sense by 
the commentary, and which enters into the title of the chapter, graha- 
yutyadkikdra, does not occur anywhere in the text The word which 
we translate "encounter," yuddha, means literally "war, conflict." 
Verses 18-20, and verse 22, below, give distinctive definitions of some 
of the different kinds of encounter and conjunction. 

2. When the longitude of the swiflpnoving planet is greater 
than that of the slow one, the conjunction (samyoga) is past ; oth- 
erwise, it is to come : this is the case when the two are moving 
eastward ; if, however, they are retrograding (vakrin), the con- 
trary is true. 

3. When the longitude of the one moving eastward is greater, 
the conjunction (samdgama) is past ; but when that of the one 
that is retrograding is greater, it is to come. Multiply the dis- 
tance in longitude of the planets, in minutes, by the minutes of 
daily motion of each, 

4. And divide the products by the difference of daily motions, 
if both are moving with direct, or both with retrograde, motion : 
if one is retrograding, divide by the sum of daily motions. 

5. The quotient, in minutes, etc., is to be subtracted when the 
conjunction is past, and added when it is to come : if the two are 
retrograding, the contrary : if one is retrograding, the quotients 
are additive and subtractive respectively. 

6. Thus the two planets, situated in the zodiac, are made to be 
of equal longitude, to minutes. Divide in like manner the dis- 
tance in longitude, and a quotient is obtained which is the time, 
in days, etc. 

The object of this process is to determine where and when the two 
planets of which it is desired to calculate the conjunction will have the 
same longitude. The directions given in the text are in the main so 
clear as hardly to require explication. The longitude and the rate of 
motion of the two planets in question is supposed to have been found for 
some time not far removed from that of their conjunction. Then, in 



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308 E. Burgess, etc. t [vii. 6- 

determining whether the conjunction is past or to come, and at what dis- 
tance, in arc and in time, three separate cases require to be taken into 
account — when both are advancing, when both are retrograding, and 
when one is advancing and the other retrograding. In the two former 
cases, the planets are approaching or receding from one another by the 
difference of their daily motions ; in the latter, by the stun of their'daily 
motions. The point of conjunction will be found by the following pro- 
portion : as the daily rate at which the two are approaching or receding 
from each other is to their distance in longitude, so is the daily motion 
of, each one to the distance which it will have to move before, or which 
it lias moved since, the conjunction in longitude. The time, again, 
elapsed or to elapse between the given moment and that of the conjunc- 
tion, will be found by dividing the distance in longitude by the same divi- 
sor as was used in the other part of the process, namely the daily rate 
of approach or separation of the two planets. 

The only other matter which seems to call for more special explana- 
tion than is to be found in the text is, at what moment the process of 
calculation, as thus conducted, shall commence. If a time be fixed 
upon which is too far removed — as, for instance, by an interval of sev- 
eral days — from the moment of actual conjunction, the rate of motion 
of the two planets will be liable to change in the mean time so much as 
altogether to vitiate the coneectness of the calculation. It is probable 
that, as in the calculation of to eclipse (see above, note to iv. 7-8), we arc 
supposed, before entering upon the particular process which is the sub- 
ject of this passage, to have ascertained, by previous tentative calcula- 
tions, the midnight next preceding or following the conjunction, and t«> 
have determined for that time the longitudes and rates of motion of the 
two planets. If so, the operation will give, without farther repetition, 
results having the desired degree of accuracy. The commentary, it may 
be remarked, gives us no light upon this point, as it gave us none in the 
case of the eclipse. 

We have not, however, thus ascertained the time and place of the 
conjunction. This, to the Hindu apprehension, takes place, not when 
the two planets are upon the same secondary to the ecliptic, but when 
they are upon the same secondary to the prime vertical, or upon the 
same circle passing through the north and south points of the horizon. 
Upon such a circle two stars rise and set simultaneously ; upon such a 
one they together pass the meridian : such a line, then, determines 
approximately their relative height above the horizon, each upon its own 
circle of daily revolution. We have also seen above, when considering 
the deflection (valana — see iv. 24-25), that a secondary to the prime ver- 
tical is regarded as determining the north and south directions upon the 
starry concave. To ascertain what will be the place of each planet upon 
the ecliptic when referred to it by such a circle is the object of the fol- 
lowing processes. 

7. Having calculated the measure of the day and night, and 
likewise the latitude {vilcshepa) y in minutes ; having determined 
the meridian-distance \nata) and altitude (unnata), in time, accord- 
ing to the corresponding orient ecliptic-point Qagna) — 



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vii. 12.] Surya-Sidali&nta. 809 

8. Multiply the latitude by the equinoctial shadow, and divi«?fc 
by twelve % tie quotient multiply by the meridian-distance in . 
nadis, and divide by the corresponding half-day: 

9. The result, when latitude is north, is subtract! ve in th* 
eastern hemisphere, and additive in the western; when latitude 
is south, on the other hand, it is additive in the eastern hemi- 
sphere, and likewise subtractive in the western. 

10. Multiply the minutes of latitude by the degrees of declin- 
ation of the position of the planet increased by three signs : the 
result, in seconds (vikald), is additive or subtractive, according as 
declination and latitude are of unlike or like direction. 

11. In calculating the conjunction {yoga) of a planet and an . 
asterism (nakshatra), in determining the setting and rising of a 
planet, and in finding the elevation of the mooirs cusps, this ope- 
ration for apparent longitude (drkkarman) is first prescribed. 

12. Calculate again the longitudes of the two planets for the 
determined time, and from these their latitudes : when the latter 
are of the same direction, take their difference ; otherwise, their 
sum : the result is the interval of the planets. 

The whole operation for determining the point on the ecliptic to 
which a planet, having a given latitude, will be referred by a secondary 
to the prime vertical, is called its drkkarman. Both parts of this com- 
pound we have had before — the latter, signifying " operation, process of 
calculation," in ii. 37, 42, etc. — for the former, see the notes to iii. 28- 
34, and v. 5-6 : here we are to understand it as signifying the " appar- 
ent longitude" of a planet, when referred to the ecliptic in the manner 
stated, as distinguished from its true or actual longitude, reckoned in the 
usual way: we accordingly translate the whole term, as in verse 11, 
u operation for apparent longitude." The operation, like the somewhat 
analogous one by which the ecliptic-deflection (valana) is determined 
(see above, iv. 24-25), consists of two separate processes, which receive 
in the commentary distinct names, corresponding with those applied to 
the two parts of the process for calculating the deflection. The whole 
subject may be illustrated by reference to the next figure (Fig. 28). This 
represents the projection of a part of the sphere upon a horizontal plane, 
N and E being the north and east points of the horizon, and Z the 
zenith. Let C L be the position of the ecliptic at the moment of con- 
junction in longitude, C being the orient ecliptic-point (logna) ; and let 
M be the point at which the conjunction in longitude of the two planets 
S and V, each upon its parallel of celestial latitude, c I and d V, and hav- 
ing latitude equal to S M and V M respectively, will take place. Through 
V and S draw secondaries to the prime vertical, N V and N S, meeting 
the ecliptic in v and s : these latter are the points of apparent longitude 
of the two planets, which are still removed from a true conjunction by 
the distance v s : in order to the ascertainment of the time of that true 
conjunction, it is desired to know the positions of v and *, or their re- 
spective distances from M. From F, the pole of the equator, draw also 
circles through the two planets, meeting the ecliptic in *' and v' : then, 
VOL. vi. 40 



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310 



E. Burg>:s$, etc.. 



[viL 12. 



Fijr. 28. 



in order to find M 8, we ascertain the values of * *' and M *' ; and, in 
like nmuuer, to find Mr, we ascertain the values of v v' and M v'. Now 

at the equator, or in a right 
sphere, the circles N S and 
P S would coincide, and 
the distance * ** disappear : 
hence, the amount of * »' 
being dependent upon the 
latitude latefta) : of the ob- 
serve^ K F r the process hy 
which it k calculated; is 
called the "operation For 
latitude" (akshudrkkarman, 
or else Aksha drJefatrman). 
Again, if P and P* were fhe 
same point, or if the eclip- 
tic and equator coincided* 
PSand JP'S would' coin- 
cide* and M.V would disap- 
pear; hence the process of 
calculation of M *' is called 
the " operation for ecliptic- 
deviation" (ayanadrkkar- 
nian, or dyana drkkarman). 
The latter of the two pro- 
cesses, although stated after 
the other in the test, is the 
one first explained by the 
commentary : we will also, 
as in the case of the deflec- 
tion (note to iv. 24-25), 
give to it our first attention. 
, ' The point «', to whieh the planet is referred by a circle passing through 
-the pole P, is styled by the commentary syowayroAa, " the planet's lon- 
gitxide ,aa corrected .^ distance MV> which 

it'is.destrjad to A*certttm;is called <iyanafet/4*» u the correction, in sua- 
irfces^iibrBnlipticideviatiQn." Instead* however, of finding M/, the pro- 
essrteaghte in the text finds M A the corresponding distance on the eir- 
xsle of daily revolution* D R> of the point M — wWeh is then assumed 
eqnaLto Ms'. The proportion: upojt which ihe rule, as stated in ve»e 
iO^ is. ultimately founded, is 

& : sin M S t : : M S : M t 
the triangle M S t, which is always very small, being treated as if It 
were a plane triangle, right-angled at t. But now also, as the latitude 
M S is always a small quantity, the angle P S P' may be treated as if 
equal to PMP' (not drawn in the figure); and this angle is, as was 
shown in connection with iv. 24-25, the deflection of the ecliptic from 
the equator (dyana valana) at M, which is regarded as equal to the 
declination of the point 90° in advance of M : this point, for conven- 
ience's sake, we will call M'. Our proportion becomes, then 



r 


\ 


K 

1 




r* — — __ J J 1 wr^~^i? 


R 


t 7 s A 





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vii. 12,] .SV: : ra-Siddhanta. 811 

. R;siadeclM'::MS;ai* 
all the quantities which it contains being in terms of minuter To bring 
this proportion, now, to the form in which it appears in the text, it is 
made to undergo a most fantastic and unscientific series of alterations. 
The greatest declination (ii. 28) being 24°, and its sine I397 f , which is 
nearly fifty-eight times twenty-four — since 58X24=1392 — it is assumed 
that fifty-height times the number of degrees in any given arc of declina- 
tibn will be equal to the number of minutes in the sine of that arc. 
Again, the value of radius, 8438', admits of being roughly divided into 
the two factors fifty-eight and sixty — since 58 X 60=3480. Substitut- 
ing 1 , then, these values in the proportion as stated, we have 

58X60 : 58Xdecl. M' in degr. : : latitude in min. : M t 
Cancelling, again, the common factor in the first two terms, and trans 
ferring the factor 60 to the fourth term, we obtain finally 

1 : decL M' in degr. : : latitude in min. : M t x 00 
(hat is to say, if the latitude of the planet* in minutes, be multiplied by 
the declination, m degrees, of a point 90° in advance of the planet, the 
result will be a quantity which, after being divided by sixty, or reduced 
from seconds to minutes, is to be accepted as the required interval on 
the ecliptic between the real place of the planet and the point to which 
it is referred by a secondary to the equator. 

This explanation of the role is the one given by the commentator, 
nor are we able to see that it admits of any other. The reduction of 
the original proportion to its final form is a process to which we have 
heretofore found no parallel, and which appears equally absurd and 
uncalled for. That MHb taken as equivalent to M *' has, as will appear 
from a consideration of the next process, a certain propriety. 

The value of the arc M *' being thus found, the question arises, in 
which direction it shall be measured from M. This depends upon the 
position of M with reference to the solstitial colure. At the colure, the 
lines PS and PS coincide, so that, whatever be the latitude of a planet, 
it will, by a secondary to the equator, be referred to the ecliptic at its 
true point of longitude. From tne winter solstice onward to the summer 
solstice, or when the point M is upon the sun's northward path (uitarA- 
?tt*a), a planet having north latitude will bo referred backward on the 
ecliptic by a circle from the pole, and a planet having south latitude will 
he referred forward. If M, on the other hand, be upon the sun's south- 
ward path (dakskm&yana), a planet having north latitude at that point 
-will be referred forward, and one having south latitude backward: this 
is the case illustrated by the figure. The statement of the text virtually 
agrees with this, it being evident that, when M is on the northward 
v -pa$h, the declination of Sic point 90° in advance of it will be north ^ and 
the contrary,, . , \ 

\ We come now to consider the other part of the operation, "or the 
4fiiha, drklcarman, which forms the subject of verses 7-9. As the first 
-step, we are directed to ascertain the day and the night respectively of 
_the point of the ecliptic at which the two planets are in conjunction' In 
longitude, for the purpose of determining also its distance in time from 
the horizon and from the meridian. This is accomplished, as follows. 



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812 2£ Burgess, etc., [viL 12. 

Having the longitude of the point in question (M in the last figure), 
we calculate (by ii. 28) its declination, which gives us (by ii. 60) the 
radius of its diurnal circle, and (by ii. 61) its ascensional difference; 
whence, again, is derived (by ii. 62-68) the length of its day and night 
Again, having the time of conjunction at M, we easily calculate the 
sun's longitude at the moment, and this and the time together give us 
(by iii. 46-48) the longitude of C, the orient ecliptic-point : then (by 
iii. 50) we ascertain directly the difference between the time when M rose 
and that when C rises, which is the altitude in time (unnata) of M : the 
difference between this and the half-day is the meridian-distance in time 
(nata) of the same point If the conjunction takes place when M is 
below the horizon, or during its night, its distance from the horizon and 
from the inferior meridian is determined in like manner. 

The direct object of this part of the general process being to find the 
value of 8 *', we note first that that distance is evidently greatest at the 
horizon ; farther, that it disappears at the meridian, where the lines PS 
and N S coincide. If, then, it is argued, its value at the horizon can be 
ascertained, we may assume it to vary as the distance from the meridian. 
The accompanying figure (Fig. 29) will illustrate the method by which 
it is attempted to calculate s s' at the horizon. Suppose the planet S, 
pig 29. * being removed in latitude to the distance 

M S from M, the point of the ecliptic 
which determines its longitude, to be upon 
the horizon, and let *', as before, be the 
point to which it is referred by a circle 
I .from the north pole : it is desired to deter- 
mine the value of * *'. Let D R be the 
circle of diurnal revolution of the point 
M, meeting S *' in f, and the horizon in w : 
S 1 20 may be regarded as a plane right-angled triangle, having its angles 
at S and w respectively equal to the observer's latitude and co-latitude. 
In that triangle, to find the value of t ta, we should make the proportion 

cos t S w : sin t S w : : t S : t w 
Now the first of these ratios, that of the cosine to the sine of latitude, 
is (see above, iii. 17) the same with that of the gnomon to the equinoc- 
tial shadow : again, as the difference of M t and M *' was in the pre- 
ceding process neglected, so here the difference of SM and Sf; and 
finally, t w, the true result of the process, is accepted as the equivalent 
of s* s f the distance sought The proportion then becomes 

gnom. : eq. shad. : : latitude : required dist at horizon 
The value of the required distance at the horizon having been thus 
ascertained, its value at any given altitude is, as pointed out above, deter- 
mined by a proportion, as follows : as the planet's distance in time from 
the meridian when updn the horizon is to the value of this correction at 
the horizon, so is any given distance from the meridian (nata) to the 
value at that distance ; or 

half-day : mer.-dist. in time : : result of last proportion : required distance 
The direction in which the distance thus found is to be reckoned, start- 
ing in each case from the dyana graha, or place of the planet on the 




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tii. 12.] StLryaSiddh&nta. 813 

ecliptic as determined by a secondary to the equator, which was ascer- 
tained by the preceding process, is evidently as the text states it in verse 
9. In the eastern hemisphere, which is the case illustrated by the figure, 
*'$ is additive to the longitude of *', while v'v is subtractive from the 
longitude of v* : in the western hemisphere, the contrary would be the 
case. The final result thus arrived at is the longitude of the two points 
* and v, to which S and V are referred by the circles N S and N V, 
drawn through them from the north and south points of the horizon. 

The many inaccuracies involved in these calculations are too palpable 
to require pointing out in detail. The whole operation is a roughly 
approximative one, of which the errors are kept within limits, and the 
result rendered sufficiently correct, only by the general minuteness of 
the quantity entering into it as its main element — namely, the latitude 
of a planet— and by the absence of any severe practical test of its accu- 
racy. It may be remarked that the commentary is well aware o£ and 
points out, most of the errors of the processes, excusing them by its 
stereotyped plea of their insignificance, and the merciful disposition of 
the divine author of the treatise. 

Having thus obtained s and v, the apparent longitudes of the two 
planets at the time when their true longitude is M, the question arises, 
how we shall determine the time of apparent conjunction. Upon this 
point the text gives us no light at all : according to the commentary, we 
are to repeat the process prescribed in verses 2-6 above, determining, 
from a consideration of the rate and direction of motion of the planets 
in connection with their new places, whether the conjunction sought for 
is past or to come, and then ascertaining, by dividing the distance v s 
by their daily rate of approach or recession, the time of the conjunction. 
It is evident, however, that one of the elements of the process of correc- 
tion for latitude (akshadrkkarman), namely the meridian-distance, is 
changing so rapidly, as compared with the slow motion of the planets in 
their orbits, that such a process could not yield results at all approaching 
to accuracy : it also appears that two slow-moving planets might have 
more than one, and even several apparent conjunctions on successive 
days, at different times in the day, being found to stand together upon 
the same secondary to the prime vertical at different altitudes. We 
do not see how this difficulty is met by anything in the text or in 
the commentary. The text, assuming the moment of apparent conjunc- 
tion to have been, by whatever method, already determined, goes on to 
direct us, in verse 12, to calculate anew, for that moment, the latitudes 
of the two planets, in order to obtain their distance from one another. 
Here, again, is a slight inaccuracy : the interval between the two, meas- 
ured upon a secondary to the prime vertical, is not precisely equal to 
the sum or difference of their latitudes, which are measured upon second- 
aries to the ecliptic. The ascertainment of this interval is necessary, in 
order to determine the name and character of the conjunction, as will 
appear farther on (w. 18-20, 22). 

The cases mentioned in verse 11, in which, as well as in calculating 
the conjunctions of two planets with one another, this operation for 
apparent longitude (drkkarman) needs to be performed, are the subjects 
of the three following chapters. 



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314 



M Burgess, etc^ 



[vii. 13- 



. 13. The diameters upon the . moon's orbit :of Morn, Saturn, 
Mercury) and Jupiter, are declared to be tfairty, iiusreaBed suc- 
cessively by half the half ; that t>f : Venu$ is aifcty. ••...-.. 
14. These, divided by the sum of Tadius and the fourth hypoth- 
enuse, multiplied by two, and again multiplied by tadius, are the 
respective corrected (sphuta) diameters : divided by fifteen, they 
are the measures (mdna) in minutes. ' 

.. We hare seen above, in connection with the calculation of eclipses 
(iv. 2-5), that the diameters of the sun, moon, and shadow had to be 
redaced r for measurement in minutes, to the moon's mean distance, at 
which fifteen yojanaa make a minute of arc. Here we find the dimen- 
sions of the fire lesser planets, when at their mean distances from the 
earth, stated only in the form of the portion of the moon's mean orbit 
covered by them, their absolute size being left undetermined. We add 
them below, in a tabular form, both in yojanas and as reduced to min- 
utes, appending also the corresponding estimates of Tycho Brahe (which 
we take from Ddanibre), and the true apparent diameters of the plan-! 
eta, as seen from the earth at their greatest and least distances* 

Apparent Diameters of the Planets, according to the Siirya-Siddhdnla t 
to Tycho Brahe, and to Modern Science, 



Planet. 


Surya-Siddhanta : 


Tycho 


Moderns : 


in yojanaf . 


in arc. 


Brahe. 


leant. greatest 


Mars, 


3o 


2' 


i' 40" 


4" 


* 7 " 


Saturn, 


37* 


a' 3o" 


i' 5o" 


i5" 


ai" 


Mercury, 


45 


3' 


a' io" 


4" 


IS" 


Jupiter, 


5a* 


3' 3o" 


a' 45" 


3c/' 


49" 


Venus, 


6o 


4' 


3' i5" 


9"'i 


i> i4" 



. This table shows how greatly exaggerated are wont to be any deter* 
ruinations of the magnitude of the planetary orbs made by the unas- 
sisted eye alone. This effect is due to the well-known phenomenon of 
the irradiation, which increases the apparent size of a- brilliant body 
when seen at some distance. It will be noticed that the Hindu esti- 
mates do not greatly exceed those of Tycho, the most noted and accu- 
rate of astronomical observers prior to the invention of the telescope. 
In respect to order of magnitude they, entirely agree, and both accord 
with tue relative apparent size of the planets, except that to Mercury 
and Venus, whose proportional brilliancy, from their nearness to the 
sun, is greater, is assigned too high a rank. Tycho also established a 
scale of apparent diameters for the fixed stars,. varying from 2', for the 
first magnitude, down to 20", for the sixth. We do not find that 
Ptolemy made any similar estimates, either for planets or for fixed stars. 
The Hindus, however, push their empiricism one step farther, gravely 
laying down a rule by which, from these mean values, the true values 
of the apparent diameters at any given time maybe found. The funda- 
mental proportion is, of course, 

true dist, ; meaji disk : : mean app. diam. : true app. 4iam. 



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vh\15.] Sfoya'Siddh&nta, 815 

The second term of this proportion is represented by radios : for the 
first we have, according to the translation given, one half the sum of 
radius and the fourth nypothenuse, by which is meant the " variable 
hypothenuse" (cala karna) found in the course of the fourth, <Jr last, 
process for finding the true place of the placet /see above, il 43-45). 
The term, however (tticatuhhirnu), which is translated " radius and the 
fourth hypothenuse" is much more naturally rendered " third and fourth 
hypothenuses" ; and the latter interpretation is also mentioned by the 
commentator aa* one handed down hy tradition, (ttimpraddyika) : but, 
he add% owing to the iaet that the length of the hypothenuse is not 
ralenkted. in* the third process, that for finding. d&naky the equation of 
the centre (mundwk<xhnan), and that that hypotdienuse cannot therefore 
be referred to here aa icBom^ modern interpreters understand the first 
member of the compound (tri) as an abbreviation for u radius" (tryyA), 
and translate it accordingly. We mast confess thai the other interpre- 
tation seems to ns to be powerfully supported by both the letter of the 
text and the reason of the matter. The substitution of tri for trijy& in 
such a. connection is quite too violent to be borne, nor do we see why 
half the sum of radius and the fourth hypothenuse should be taken as 
representing the planet's true distance, rather than the fourth hypothe- 
nnse alone, which was employed (see above, ii. 56-58) in calculating the 
latitude of the planets. On the other hand, there is reason for adopt- 
ing, as the relative value of a planet's true distance, the average, or half 
the sum, of the third hypothenuse, or the planet's distance as affected 
by the eccentricity of its orbit, and the fourth, or its distance as affected 
by the motion of the earth in her orbit. There seems to as good 
reason, therefore, to suspect that verse 14 — and with it, probably, also 
verse 13— is an intrusion into the Snrya-Siddh&nta from some other sys- 
tem, which did not make the grossly erroneous assumption, pointed out 
under ii. 30, of the equality of the sine of anomaly in the epicycle 
(bhujajydphala) with the sine of the equation, but in which the hypoth- 
enuse and the sine of the equation were duly calculated in the process 
for finding the equation of the apsis [mandakarman^ as well as in that 
for finding the equation of the conjunction {pighrahirman). 

.. 15. Exhibit^ upon the shadow-ground, the planet at the ex» 
tremity of its shadow reversed : it is viewed at the apex of the 
gBLomoH ia.its mi^rpr; 

As a practical test of the accuracy of his calculations, or as a con- 
vincing proof to the pupil or other person of his knowledge and skill, 
the teacher is here directed to set up a gnomon upon ground properly 
prepared for exhibiting the shadow, and to calculate and lay off from the 
base of the gnomon, but in the opposite to the true direction, the shad- 
ow which a planet would cast at a given time ; upon placing, then, a 
horizontal mirror at the extremity of the shadow, the reflected image of 
the planet's disk will be seen in it at the given time by an eye placed at 
the apex of the gnomon. The principle of the experiment is clearly 
correct, and the rules and processes taught in the second and third chap- 
ters afford the means of carrying it out, since from them the shadow 
which any star would cast, had it light enough, may be as readily deter- 



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*16 j£ Butg&8 f fte; 9 ;„ [vim** 

minejias. that, which the sun actually cast*. As fop case of precisely this 
character has hitherto been presented, we will briefly indicate the course 
qi the calculation.- The day and night of the planet, and its distance 
from the meridian, or its hour-angle, are found in the same manner as in 
the process previously examined Jp. 312, above), excepting that here the 
planet's latitude, and its declination as affected by latitude, most be cat* 
culated, by ii. 5Q-58 ; and then the bjom v angle and the ascensional differ- 
ence, by hi. 34-36, give the length of the shadow at the given time, 
together with that of its hypothenuse. The question would next be in 
what direction to lay off the shadow from the . base of the gnomon.' 
This is accomplished by means of the base (bhvja) of the shadow, or its: 
value when projected on a north and south line* From the declination 
is found, by iiL 20-22, the length of the noon-shadow and its hypothec 
rinse, and from the latter, with the deolination, comes, by iii. 22-23, ther 
measure of .amplitude (agr&) of the given shadow ; whence, by tir. 28~ 
25, is derived its base. Having thus both its length and the distance 
of its extremity from an east and west line running through the base of 
the gnomon, we lay it off without difficulty. 

16. Take two gnomons, five cubits (hasta) in height, stationed" 
according to the variation of direction, separated by the inter- 
val of the two planets, and buried at the base one cubit. 

17. Then fix the two hypothenuses of the shadow, passing 
from the extremity of the Shadow through the apex of each 
gnompn : and, to a person situated at the point of union of the 
extremities of the shadow and hypothenuse, exhibit 

-IB; The two planets in the sky, situated at the apex each of 
its own gnomon, and arrived at a coincidence of observed place 

r This is. a proceeding of much the same character with that which 
forms the subject of. the preceding passage. In order to make appro- ' 
hetisible, by observation, the conjunction of two planets, as calculated by , 
the methods of this chapter, two gnomons, of about the height of a 
mail, are set up. At what distance and direction from one another they 
are to be fixed is not clearly shown. The commentator interprets the 
expression 'HntemTof the two planets" (vi 14), to mean their distance in , 
minutes on the secondary to the prime vertical, as ascertained according 
to verse 12, above, reduced to digits 'by the method taught in iv. 26 ; 
while, by u according to the variation of direction," he would understand 
merely? in the direction from the observer of the hemisphere in which the 
planets at the moment of conjunction are situated. The latter phrase,, 
however, as thaa explained, seems utterly nugatory ; nor do we see of 
what use it would be to make the north and south interval of the bases of 
the gnomons, in digits, correspond with that of the planets in minutes. 
We do not think it would be difficult to understand the directions given 
in the text as meaning, in effect, that the two gnomons should be so sta- 
tioned as to cast their shadows to the same point : it would be easy to 
do this, since, at the time in question, the extremities of two shadows 
cast from one gnomon by the two stars would be in the same north and 



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▼U.2&] S&rya-Siddhdnta. 817 

sooth line, and it would only bo necessary to set the second gnomon as 
far south of the first as the end of the shadow cast by the southern star 
was. north of that oast by the other. Then, if a hole were sunk in the 
ground at the point of intersection of the two shadows, and a person 
enabled to place his eye there, he would, at the proper moment, see both 
the planets with the same glance, and each at the apex of its own gnomon. 

In the eighteenth verse also we have ventured to disregard the author- 
ity of the commentator : he translates the words drktulyat&m it As 
"come within the sphere of sight " while we understand by drktulyaid, 
as in other cases (ii. 14, iii. 11), the coincidence between observed and 
computed position. 

Such passages as this and the preceding are not without interest and 
value, as exhibiting the rudeness of the Hindu methods of observation, 
and also as showing the unimportant and merely illustrative part which 
observation was meant to play in their developed system of astronomy. 

18. , . . When there is contact of the stars, it is styled ".de- 
piction" (ullekha) ; when there is separation, "division" (blieda) ; 

19. An encounter (jjuddha) is called " ray-obliteration" (angu- 
vimarda) when there is mutual mingling of rays: when the inter* 
val is less than a degree, the encounter is named " dexter" (apa- 
savya) — i£ in this case, one be faint (anu). 

20. If the interval be more than a degree, it is "conjunction" 
(samdjama), if both are endued with power (bala). One that is 
vanquished (Jita) in a dexter encounter (apasavya yuddha) } one 
that is covered, faint (anu), destitute of brilliancy, 

21. One that is rough, colorless, struck down (vidhvasta), situ- 
ated to the south, is utterly vanquished (vijita). One situated to 
the north, having brilliancy, large, is victor (Jayin) — and even in 
the south, if powerful (balin). 

22. Even when closely approached, if both are brilliant, it is 
u conjunction" (samdgama) : if the two are very small, and struck 
down, it is "front" (kAia) and "conflict" (vigraha\ respectively. 

23. Venus is generally victor, whether situated to the north or 
to the south. . . . 

In this passage, as later in a whole chapter (chap, xi), we quit the 
proper domain of astronomy, and trench upon that ot astrology. How- 
ever intimately connected the two sciences may be in practice, they are* 
in general, kept distinct in treatment — the Siddhantas, or astronomical 
text-books, furnishing, as in the present instance, only the scientific basis, 
the data and methods of calculation of the positions of the heavenly 
bodies, their eclipses, conjunctions, risings and settings, and the like, 
while the Sanhitas, Jatakas, Tajikas, etc., the astrological treatises, make 
the superstitious applications of the science to the explanation of the 
planetary influences, and their determination of human fates. Thus the 
celebrated astronomer, Yaraha-mihira, besides his astronomies, com- 
posed separate astrological works, which are still extant, while the for- 
mer have become lost. It is by no means impossible that these verses 
may be an interpolation into the original text of the Surya-Siddhanta. 
They form only a disconnected fragment : it is not to be supposed that 
vol. vi. 41 



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they contain a complete statement and definition of all the different 
kinds of conjunction recognized and distinguished by technical appella- 
tions ; nor do they fully set forth the circumstances which determine the 
result of a hostile " encounter" between two planets : while a detailed 
explanation of some of the distinctions indicated — as, for instance, when 
a. planet is ^powerful" or the contrary— could not be given without enter- 
ing quite deeply into the subject of the Hindu astrology. This we do 
not regard ourselves as called upon to do here : indeed, it would not be 
possible to accomplish it satisfactorily without aid from original sources 
which are not accessible to us. We shall content ourselves with follow- 
ing the example of the commentator, who explains simply the sense and 
connection of the verses, as given in our translation, citing one or two 
parallel passages from works, of kindred subject We would only point 
out farther that it has been shown in the most satisfactory manner (as by 
Whiah, in Trans. Lit Soc. Madras, 1 82 V ; Weber, in his Indische Studien, 
ii. 236 etc.) that the older Hindu science of astrology, as represented by 
Varaha-mihira and others, reposes entirely upon the Greek, as its later 
forms depend also*, hi part, upon the' Arab; the latter connection being 
indicated even m the common fitte tof tfte more modern treatises, t&jika f 
which comes from the Persian ids^ u Arab." Weber gives (Ind. Stud. 
ii. 2*?7 etc.) a translation of a passage from Var&ha-mihira's lesser treat- 
ise, which states in part the tiircumstances determining the "power" of a 
planet m different situations, absolute or relative: partial explanations 
upon the same subject famished to the translator in India by his native 
assistant, agree with these, and both acoord closely with the teachings 
Of the Tetrabiblos, the astrological work attributed to Ptolemy. 

' 23. . . . Perform in like manner the calculation of the con- 
junction (samyoga) of the planets with the moon. 
. This is all that the treatise says respecting the conjunction of the 
moon with the lesser planets : of the phenonmoa, sometimes so striking, 
of the occultation of the latter by the forme*, it takes no especial notice. 
The commentator cites an additional half-verse as sometimes included in 
the chapter, to the effect jthat, in calculating a conjunction, the moon's 
latitude is to be reckoned as corrected by Iter parallax in latitude (ava- 
qafi) 9 but rejects it, as making the chapter over-full, and as being super- 
fluous, since the nature of the case determines the application here of 
the general rules for parallax presented in the fifth chapter. Of any 

Sarallax of the planets themselves nothing is said : of course, to calcu- 
li* the moon's parallax by the methods as already given is, in effect, to 
attribute to them all, a horizontal parallax of the same value with that 
assigned to the sun, or about 4'. 

The final verse of the chapter is a caveat against the supposition that, 
when a " conjunction", of two planets is spoken of, anything more is 
meant than that they appear tp approach one another ; while neverthe- 
less, this apparent approach requires to fee treated of, on account of its 
influence uponhumap fates. . . 

24. Unto tho good and. evil fortune. of men is this system set 
fotth : the planets move* on: upoa tbei^own paths, approaching 
one another at * distance. 



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viii.l.] 



8&rya-$iddhd?nta. 



$1§ 



CHAPTER VIII. 

OF THE ASTEBISMS. 

Coxtemts :— 1-9, positions of the asterisms ; 10-12, of certain fixed stars ; 12, direc- 
tion to test by observation the accuracy of these position! ; 18, splitting of 
Rohinfs wain; 14-15, how to determine the conjunction of a planet with an 
aaterism ; 16-19, which is the junction-star in each asterism ; 20-21, positions of 
other fixed stars. 

1. Now are set forth the positions of the asterisms (bha), in 
minutes. If the share of each one, then, be multiplied by ten, 
and increased by the minutes in the portions (bhoga) of the past 
asterisms (dhishnya), the result will be the polar longitudes 
(dhruva). 

The proper title of this chapter ia nakikaJmgrahtyutyadhikAr*, " ehap- 
ter of the conjunction of asterisms and planets," but the subject of con- 
junction occupies but a small space in it, being limited to a direction 
(vv. 14-15) to apply, with the necessary modifications, the methods 
taught in the preceding chapter. The chapter is mainly occupied with 
such a definition of the positions of the asterisms — to which are added 
also those of a few of the more prominent among the fixed stars — as is 
necessary in order to render their conjunctions capable of being calculated. 
Before proceeding to give the passage which states the positions <pf 
the asterisms, we will explain the manner in which these .are defined. In 
the accompanying figure (Fig. 30), let £ L represent the equator, and C 
_. • L the ecliptic, r and P' being their respec- 

*" ' tive poles. Let S be the position of any 

given star, and through it draw the circle 
of declination P 8 a. Then a is the point 
on the ecliptic of which the distance from 
the first of Aries and from the star respec- 
tively are here given as its longitude and 
latitude. So far as the latitude is con- 
cerned, this is not unaccordant with the 
usage of the treatise hitherto. Latitude 
(vikskepa, " disjection") is the amount by 
which any body is removed from the 
declination which it* ought to hare— that 
is, from the point of the ecliptic which it 
ought "to occupy— declination (br&nli, opo- 
krama) being always, according to the 
Hindu understanding: of the term, in the 
ecliptic ItseMl In the case of a^nlariet, 
whose proper path is in the ecliptic, the 
point of that circle which it ought to occu- 
py is determined fer. kacalcQlated longif: 
tude : in the case of a fixed star, whose only motion if about ^be pole of 
the heavens, its point of declination is that to which it is referred by a 




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320 , E. Buiym, etc> / [viii 1^ 

citdethrangbtbjtple. Tfrg, io tfra figpi* ft»Jfrihiltn» (kr4n*k)mt : 
8 would be c «, or the distant* of a frapi 4h» *qaat*r il e«> Jta latitat*: 
{yilc$hepa) it a S, or itB di^nceftem *. . Wehwo, accordingly, Aer 
same term used here as Wore, To designate the ppsitie* hi longitude.: 
of a, on the other hand, we have a new ternx dkrwo* or*as below, (va^: 
12, ihX, dhruvaka. This oomes from the.adjective. dhruv* r "fixed, im*.: 
movable," by which the poles of the heaven (see below, X&.43) are desig- 
nated ; and, if we do not mistake its . application, it indicates, as here 
eraploved, the longitude of a star aa referred to the ecliptic by a. circle:: 
from the pole. We venture, then, to translate it by "polar bngitode,"*; 
as we also render vikshepa, in this connection, by " polar latitude," at: 
being desirable to have for these quantities distinctive Barnes, akin? 
with one another. Colebrooke employs u apparent longitude and lafcU : 
tude," which are objectionable, as being more properly applied to the 
results of the process taught in the last chapter (vv. 7-10).. . 

The mode of statement of the polar longitudes is highly artificialand - 
arbitrary : a number is mentioned which, when multiplied by ten, will : 
give the position of each asterism, in minutes, in its own "portion" 
(M^ra), or jure of 13° 20' in the ecliptic (see ii. 64). — , 

This passage presents a name for the asterisms, dhisknya, which h*»< 
not occurred before ; it is found once more below, in xi. 21. ' " j__ ! 

. 2. Forty-eighty forty, sixty-five, fifty-seven, fifty -eight*, four, ; 
aeventy-eight, geventy-six, fourteen, 

8. Fifty-four, sixty -four, fifty, sixty, forty, ,eeffenty-foar r 4rty* ! 
enty-eight, sixty 'four, :.-..; 

4. Fourteen, «ix, lour: tTttara*Ashfidhfi, (vdifva) 4m at the 
middle of the portion (bhoga) of Parv*Afch«db« (dpya) ; Abhi- 
jit, likewise,, is at the end of Piirva-Asbfidha; the petition j>f 
^ravasa is at the end of Uttara- Ashfidhfi ?•'"•' 
^ 6. QravishthiL on the other hand, is at the point of connec- ; 
tion of the third and fourth quarters (pada) of 9 r »va?La : then, j 
in their own portions, eighty, thirty-six, twenty-two, ; 

6. Seventy^nine. Now their respective latitudes, reckoned j 
from the point of declination (apakrama) of each : ten, twelve,. : 
five, north; south, five, ten,, nine; 

7. North, six ; nothing; south, seven ; north, nothing, twelve, I 
thirteen ; south, eleven, two ; then thirty-seven, north ; 

8. South, one and a half, three, four, nine, five and a half; five ; ' 
north, also, sixty, thirty, and also thirty-six ; j 

9. South, half a degree; twenty*four, north, twenty-six degrees; j 
nothing— for A$vinf (cfcwra), etc., in succession. 

The text here assumes that the names of the asterisms, and the order - 
of their succession, are so familiarly known as to render it unnecessary 
to rehearse them. It has been already noticed (see above, i. 48*51, 56, . 
56-58, etc.) that a aiinilar assumption was .made as regards the nameaT. 
and«tfocession of the montbs, Signa of'the sodiac^ years r of Jupiter's' ] 
cycle, and the -Mice. "Many of the asterisms have more than one-appel- ■! 
latiofe t we present in the annexed table those by which thejnsre moro^ ( 



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▼nifcj. 



SH^yn^mH&nia; 



881- 



gbm*ra% ifrt'jtadi^tkQemt •» *«*» wilP*^ stki^ fertter ''cm/ 
Keartyalkthes* tfciWBWto*$e *&&<* Ifi ^ft«%xt»-6cc^Tftg ^ereYand* 

aBQLiuestto0ed wily by appelfsltibro derive<£f*6m lh6 names of ibe Sin-* 
ties to whom they are regarded asbelbngrag, and one (the 25&) chances? 
not to be once distinctively spoken of; We append to the names, in a* 
tabular form, the data presented in this passage; namely,, the posit joir 
ot each asterism {m*Jb&*Jr«) in the am of the ecliptic to which rt gives* 
nam^ and which is styled its "portion" (©Aoja); the resulting polar Ion-: 
gitodes, and the polar latitadee* And since it is probable (see note to 
the latter half of v. 12, below > that the latter were actually derived \>y 
calculation from true declinations and right ascensions, ascertained T)y* 
observation, we have endeavored to restore those more original data 'by ' 
cafenhtingthetn back again, according *6 the data and methods, of fills' 
$iddh&nta — the declinations by ii« 28, the right ascensions by iii. 44-48' 
—j-and we insert our results in the table, ejecting odd minutes less than 
ten,- - - • •• 

Pmtiw of tk* JutuAion-Stcfr* of the Asteritms. ** 



HV 


Nun*. 


PMki n 

ift It* 
r<miwu J 


Msr 

Loogiud«. 


'Mac 

JUoitucU. 


ftifht 


Declination. 


LrtDfUud*. 


lqf|rtftl 


•: 




„ 


o r 


o » 


/ . 


/ 


-/ 


.'. . 


-9 #> 




I 


Actio!; 


8 o 


« o 


to oN. 


. 7 3o 


*3 aoN. 


13 O. 


•*?, 


,3 


2 


Bhsrant, 


6 4o 


ao o 


13 0-<» 


r8 3o 


20 d " 


17 .3o 

.13, -0 

1 3 3o ; 


16 5o 




3 

4 


Kfttttt, 
Rohint, 


to So 
9 3o 


•".37- 3o 
49 3o 


5 oS. 


35 so 

47 ao 


19 30 * 

i3- «*; 


l^.-rO. 

44p, 


•a 


$ 


ifrgsffrshs, 


9-4o 


.63.. o 


io 0.- 


61 


.11 ao » 


4 ao 
35 4&; 




* 


A|4^» ':, 


.ft*. 


£* OP 


97.0* 


fiS4o, 


1*. o"; 


14 6 


•*:. 


T 


Pwamsu, 


"*3 .$ 


©3 «> 


>\*K 


,9$. 10. 


*> o;« 


13 * D 


r. 


8 


Pushva, 


ia 4o 


106 o 


00 


-107. .io' 


a3 M 


3 


3 20 

20 4o 


w, 


9 


Acletha, 


a^ao 


109 


7 08. 


no 3o 


i5^« 


20 




10 


Moghft, 


9 o 


139 





i3t 10 


18 ao ■* 


i5 


1 5 


:, » 


n 


P.-PhalgunS, 


xo 4o 


x44 o 


13 ON. 


146 10 


a5 5o " 


it 


10 4o 




13 


U.-Pbalgunl, 


8 ao 


i55 


i3 0" 


i56 5o 


33 50 " 


i5 


i3 So 




t* 


Oasts, 


IO o 


170 


11 oS. 


170 4o 


7 oS. 


10 


9 20 

17 4o 
j3 io 


♦, 


u 


Ota* 


6 4o 


180 


3 0" 


180 


a " 


19 .0 

i4 

11 

5 Q 

12 




i5 


SvAti, 


ia ap 


199 


37 oN. 


197 4o 


39 ao N. 




16 

17 


VifOoiay 
AnurftdhA, 


i3 o 
io 4o 


at3 

334 


r 3oS. 
3 0" 


aio 5o 
aar 5o 


i4 ao 8. 
19 20 '■ 


11 
5 




id 


Jywhths, 


a ao 


339 O 


4 0" 


aa6 5o 


21 5o* 


12 




'9 


Mais, 


I o 


341 O 


9 o 14 


338 5o 


39 5o 4 * 


i3 6 


i4 

6 36 




20 


^•AsWdW, 


o*> 


354 


5 3o« 


353 5o 


38 3o " 


6 




3! 


U.-Ashsdhs, 


• ••• 


360 


5 o w 


359 30 


a8^o u 


6 4o 


7 
14 3o 




33 


Abhijit, 


.... 


a66 4o 


60 oN. 


366 20 


36 N. 


i3 .20 




33 


Qravans, 


.... 


a8o 


3o o M 


280 5o 


6 20 4i 


10 


»o4e 

3o 4o 




*r 


pmviththA, 


.... 


390 


36 o" 


391 3o 


i3 30." 


3o 




35 


(yatubhishaj, 


i3 ao 


3so 


3o8. 


3aa 10 


i5 4cr 8. 


v6 


fro 




35 


fVBhadranad* 


6 o 


336 


a4 oN. 


3a8 ID* 


io 5o N; 


-}l; r .jOt; 


: >p^0 


.'. 


af 


tr.;BMdnfpa<ft» 


3 4o 


337 


36 o« 


338-40 


T6,56^ 


- *3 : . 60; 


-M 10 




38 


fUwttf * : 


r3 io 


359 50 


- b * 


■*9. * 


' - J 


,-^'Vov 


1 7*» 


:.. 



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822 



A(vini. 



Bharant 



3. 
KrtUkl 



4. 
Rohiijii. 



5. 
Mfgagirsha. 



6. 
Ardrl 



Punarvasu. « 



8. 
Puabya. 



9. 

Aflesba. 



10t 

Magba. ' 



ir. 
P.-Phalguni. 



19. 

U.-Phalgunt 



i3. 
Hasta. 

i4. 
Citrl 



0° 



% E. Burgess, ete., 4 

Fig. 81. 



S60° 



\K. 



-1. A(TinL 



a. Bharaitf. 



Bb. 



K. 



3. Kfttika, 

4. RohiQt 



1 1 

1 



I 5, 
lie 

5. Mfgafffifca. 
I -6. ArdriL ■ 

A. 

7. 

I P. 

-7. Pnoarraao. 



-8. Puahya. 
9. AfleshA, 



A. 

1 10, 

It 

10. Magba, 

11. 
P.-Ph. 

(i.P.*PhalgtiDt 
lib 

U.-Ph. 
1 a U^Pbalguot 

is. 

|H. 

1 3. Haste. 

I 14. 

c. 



-14. Oitra. 



28. Rerati— 1 

28.1 

R.| 



37. 1 

U.-Bh 
a7.U.-Bbadrapatla- I 

36 I 

P..B1. 
26. P,-Bhadrapatla~ 

a5. fatabbisbaj- 

35.1 

fat. 

14.1 
94 



34. fraviihthi- 

<; 

a3. fraTana-l 

3'J.f 

A I 

la. Abbiiit- 
j 1 

V.- A. I 

an. U.-Ashadha— I 

ao. P.-Ashadha- 
20J 
P.-AJ 

19. Mula— I 

19 I 

m| 



161 



18. Jyesbtba- 

11 

17* Anuriltlba- 

v 

i& Vi?akba- 

J 

I 
1 5. Srati— I 

v.r 
■ 

14. CitrA— I 



[viii, 9. 



27. 
Revati. 



I *6. 
1 U.-Bhadrapt 



a5. 
P.*Bbadrap. 



Qaiabbiabaj. 



^3. 
£raTiahtha. 



22. 
gtava^a. 



21. 

U.-Ashadha. 



20. 
P.-Aabadba. 



19. 

Mula. 



i9. 
JyeabtUL 



*7- 
Anaridha. 



:i6. 



i5. 
SratL 

i4. 
Citrl 



18U Q 



uo° 



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X 



viiL 9 J SAryv-JSiddhdnta. 823 

Our calculations, it should be remarked, are founded upon the as- 
sumption, that, at the time when the observations were made of which 
our text records the' results, the vernal equinox coincided with the 

-initial point of the Hindu sidereal sphere, or with the beginning of 
the portion of the asterism Acvinl, a point 10' eastward on the ecliptic 
from the star £ Piscium : this was actually the case (see above, under i. 
27) about A. D. 560. The question how far this assumption is sup- 
ported by evidence contained in the data themselves will be considered 
later. To fill out the table, we have also added the intervals in right 
ascension and in polar longitude. 

The stars of which the text thus accurately defines the positions do 
not, in most cases, by themselves alone, constitute the asterisms (nak- 
shatra) ; they are only the principal members of the several groups of 
stars — each, in the calculation of conjunctions (yoga) between the plan- 
ets and the asterisms (see below, w. 14-15), representing its group, and 
therefore called (see below, vv. 16-19) the ."junction-star" (yogat&rd) of 
the asterism. 

It will be at once noticed that while, in a former passage (ii. 64), the 
ecliptie was divided into twenty-seven equal arcs, as portions for the aster- 
isms, we have here presented to us twe*ty*eigbt asterisms, very unequally 
distributed along the ecliptic, and at greatly varying distances from it. 
And it is a point of so much consequence, in order to the right under*: 
standing of the character and history of the whole system, to apprehend 
clearly the relation of the groups of stars to the arcs allotted to them, 
that we have prepared the -accompanying diagram (Fig. 31) in illustra- 
tion of that relation. The figure represents, in two parts, the circle of 
the ecliptic : along the central lines is marked its division into arcs of 
ten and five degrees : upon the outside *f 4hese lines it is farther divided 
into equal twenty-sevenths, or arcs of 13° 20', and upon the inside into 

. equal twenty-eighths, or. arcs o£ 12° 51$'; these being the portions 
(bhoga) of two systems of asterisms, twenty-seven and twenty-eight in 
number respectively. The stored lines which run across all the divisions 
mark the polar longitudes, as stated in the- text, of the junction-stars of 
tie asterisms. The names of the latter are set over against them, in the 
inner columns : the names of the portions in the system of twenty-seven 
are given in fall in the outer columns, and those in. the system of twenty- 
eight are also placed opposite the portions, upon the inside, in an abbre- 
viated form. 
The text nowhere expressly states which one of the twenty-eight aster- 

. isms which it recognises is, in its division of the ecliptic into only twenty* 
seven portions, left without a portion. That Abhijit, the twenty-second 
of the series, is the one thus omitted, howevet, is clearly implied in the 
statements of the fourth and fifth verses. Those statements, which have 
caused difficulty to more than one expounder of the passage, and have 

< been v&riouslv misinterpreted, are made entirely clear by supplying the 
words " asterism" and " portion" throughout, where they are to be under- 
stood, thus : " the asterism -Uttara-Ashadb a is at the middle of the por- 
tion styled Purva-Ashadha ; the asterism Abhijit, likewise, is at the end 
of the portion Purva-Ashadha ; the position of the asterism Qravana is 
at the end of the portion receiving its name from U.ttara-Ashadha ; while 



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824 E. Burgess, etc., [viii. 0. 

the asterism fravishtha is between the third and fourth quarters of the 
portion named for fravana." After this interruption to the regularity of 
correspondence of the two systems — the asterism Abhijit being left with- 
out a portion, and the portion Qravishtftft, containing no asterism — they 
go on again harmoniously together to the close. The figure illustrates 
clearly this condition of things, and shows that, if Abhijit be left out of 
account, the two systems agree so far as this — that twenty-six asterisms 
fall within the limits of portions bearing the same name, while all the 
discordances are confined to one portion of the ecliptic, that comprising 
the 20th to the 23d portions. If, on the other hand, the ecliptic be divi- 
ded into twenty-eighths, and if these be assigned as portions to the 
twenty-eight asterisms, it is seen from the figure that the discordances 
between the two systems will be very great ; that only in twelve instan- 
ces will a portion be occupied by the asterism bearing its own name, and 
by that alone ; that in sixteen cases asterisms will be found to fall within 
the limits of portions of different name ; that four portions will be left 
without any asterism at all, while four others will contain two each. 

These discordances are enough of themselves to set the whole sub- 
ject of the asterisms in a new light. Whereas it might have seemed, 
from what we have seen of it heretofore, that the system was founded 
upon a division of the ecliptic into twenty-seven equal portions, and the 
selection of a star or a constellation to mark each portion, and to be, as 
it were, its ruler, it now appears that the series of twenty-eight aster- 
isms may be something independent of, and anterior to, any division of 
the ecliptic into equal arcs, and that the one may have been only arti- 
ficially brought into connection with the other, complete harmony 
between them being altogether impossible. And this view is fully sus- 
tained by evidence derivable from outside the Hindu science of astron- 
omy, and beyond the borders of India. The Parsls, the Arabs, and the 
Chinese, are found also to be in possession of a similar system of divi- 
sion of the heavens into twenty-eight portions, marked or separated by 
as many single stars or constellations. Of the Pars! system little or 
nothing is known excepting the number and names of the divisions, 
which are given in the second chapter of the Bundehesh (see Anquetil 
du Perron's Zendavesta, etc., ii. 349). The Arab divisions are styled 
man&zil alkamar, " lunar mansions, stations of the moon," being brought 
into special connection with the moon's revolution ; they are marked, 
like the Hindu " portions," by groups of stars. The first extended com- 
parison of the Hindu asterisms and the Arab mansions was made by Sir 
William Jones, in the second volume of the Asiatic Researches, for 1790 : 
it was, however, only a rude and imperfect sketch, and led its author to 
no valuable or trustworthy conclusions. The same comparison was taken 
up later, with vastly more learning and acuteness, by Colebrooke, whose 
valuable article, published also in the Asiatic Researches, for 1807 (ix. 
323, etc. ; Essays ii. 321, etc.), has ever since remained the chief source 
of knowledge respecting the Hindu asterisms and their relation to the 
lunar mansions or the Arabs. To Anquetil (as above) is due the credit 
of the first suggestion of a coincidence between the P&rst, Hindu, and 
Chinese systems : but he did nothing more than suggest it : the origin, 
character, and use of the Chinese divisions were first established, and 



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Tin. a.] B&rr/a-SidcMnta. '826 

jEheir primitiWidentity with the Hindu asterisms demonstrated, by Biot, 
in a series of articles published in the Journal des Savants for 1840 ; and 
he has moTe recently, in the volume of the same Journal for 1859, re- 
viewed and restated his former exposition and conclusions. These wo 
shall present more fully hereafter : at present it will be enough to say 
that the Chinese divisions are equatorial, not zodiacal ; that they are 
named sieu, "mansions"; and that they are the intervals in right ascen- 
sion between certain single stare, which are also called sieu f and have the 
same title with the divisions which they introduce. We propose to pre- 
sent here a summary comparison of the Hindu, Arab, and Chinese sys- 
tems, in connection with an identification of tho stars and groups of 
stars forming the Hindu asterisms, and with the statement of such 
information respecting the latter, beyond that given in our text, as will 
best contribute to a full understanding of their character. 

The identificatijp of the asterisms is founded upon tho positions o£ 
their principal or junction-stars, as stated in the astronomical text-books, 
upon the relative places of these stars in the groups of which they form 
a part, and upon the number of stars composing each group, and the 
figure by which their arrangement is represented : in a few cases, too t 
the names themselves of the asterisms are distinctive, and assist the iden- 
tification. The number and configuration of the stars forming the groups 
are not stated in out text ; we derive them mainly from Colebrooke, 
although ourselves also having had access to, and compared, most of his 
authorities, namely the Qakalya-Sanhita-, the Muhurta-Cintamani, and 
the Ratnamala (as cited by Jones, As. Res., ii. 294). Sir William Jones, 
it may be remarked, furnishes (As. Res., ii. 293, plate) an engraved copy 
of drawings made by a native artist of the figures assigned to the aster- 
isms. For the number of stars in each group we have an additional 
authority in al-B!runi, the Arab savant of the eleventh centnry, who 
travelled in India, and studied with especial care the Hindu astronomy. 
The information furnished by him with regard to the asterisms we derive 
from Biot> in the Journal des Savants for 1845 (pp. 39-54) ; it professes 
to' be founded upon the Khanda-Kataka* of Brahmagupta. Al-Biruni 
also gives an identification of the asterisms, so far as the Hindu astrono- 
mers of his day were able to furnish it to him, which was only in part :. 
he is obliged to mark seven or eight of the series as unknown or doubt- 
ful. He speaks very slightingly of the practical acquaintance with the 
heavens possessed by the Hindus of his time, and they certainly have 
not since improved in this respect ; the modern investigators of the 
same subject, as Jones and Colebrooke, also complain of the impossi- 
bility of obtaining from the native astronomers of India satisfactory 
identifications of the asterisms and their junction-stars. The translator, 
in like manner, spent much time and effort in the attempt to derive 
such information from his native assistant, but was able to arrive at no 
results which could constitute any valuable addition to those of Cole- 
brooke. It is evident that for centuries past, as at present, the native 

* The true form of the name is not altogether certain, it being known only 
through its Arabic transcription : it seems to designate rather a chapter in a treatise 
than a complete work of its author. 
vol. vi. 43 



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826 E. Burgess, etc., [viii. 0. 

tradition has been of no decisive authority as regards the position and 
composition of the groups of stars constituting the asterisms : these 
must be determined upon the evidence of the more ancient data handed 
down in the astronomical treatises. 

In order to an exact comparison of the positions of the junction-stars 
as defined by the Hindus with those of stars contained in onr cata- 
logues, we have reduced the polar longitudes and latitudes to true longi- 
tudes and latitudes, by the following formulas (see Fig. 30) ; 

(14*008 Aa) cot ELC =c tan Sab 
Bin Sab sin 8a sss bid 86 
tan S6 cot Sabzsz sin ab 

A a being the polar longitude as stated in the text (= La + 180°), Sa 
the polar latitude, ELC the inclination of the ecliptic, S b the true lati- 
tude, and a b a quantity to be added to or subtracted from the polar Ion* 
gitude to give the true longitude. The true positions^ of the stars com- 
pared we take from Flamsteed's Catalogue Brittanicus, subtracting in 
each case 15° 42' from the longitudes there given, in order to reduce 
them to distances from the vernal equinox of A. D. 560, assumed to 
coincide with the initial point of the Hindu sphere. There is some 
discordance among the different Hindu authorities, as regards the stated 
positions of the junction-stars of the asterisms. The Qakalya-SanhitA, 
indeed, agrees in every point precisely with the Surya~Siddh&nta. But 
the Siddh&ntarQiromani often gives a somewhat different value to the 
polar longitude or latitude, or both. With it, so far as the longitude is 
concerned, exactly accord the Brahma-Siddh&nta, as reported by Cole- 
brooke, and the Khanda-Kataka, as reported by al-Blruni. The lati- 
tudes of the Brahma-Siddhanta also are virtually the same with those of 
the SiddhAnta-Qiromani, their differences never amounting, save in a 
single instance, to more than 3' : but the latitudes of the Khanda-Kataka 
often vary considerably from both. The Graha-L&ghava, the only other 
authority accessible to us, presents a series of variations of its own, inde- 
pendent of those of either of the other treatises. All these differences 
are reported by us below, in treating of each separate asterism. The 
presiding divinities of the asterisms we give upon the authority of the 
Taittirlya-Sanhita (iv. 4. 10. 1-3), the Taittiriya-Brahmana fiii. 1. 1,2, as 
cited by Weber, Zeitsch. f. d. El d. Mors., vii. 266 etc., ana Ind. Stud., 
i 90 etc.), the Muhurta-CintAmani, and Colebrooke : those of about 
half the asterisms are also indirectly given in our text, in the form of 
appellations for the asterisms derived from them. 

The names and situations of the Arab lunar stations are taken from 
Ideler's Untersuchungen fiber die Sternnamen : for the Chinese man- 
sions and their determining stars we rely solely upon the articles of 
Biot, to which we have already referred. 

It has seemed to us advisable, notwithstanding the prior treatment by 
Colebrooke of the same subject, to enter into a careful re-examination 
and identification of the Hindu asterisms, because we could not accept 
in the bulk, and without modification, the conclusions at which he arrived. 
The identifications by Ideler of the Arab mansions, more thorough and 
correct than any which had been previously made, and Biot's compari- 
son of the Chinese sieu, have placed new and valuable materials tri our 



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viii. 9.] S&rya-Siddh&nta. ZVt 

hands : and these — together with a more exact comparison than was 
attempted by Colebrooke of the positions given by the Hindus to their 
junction -stare with the data of the modern catalogues, and a new and 
independent combination of the various materials which he himself fur- 
nishes — while they have led us to accept the greater number of his iden- 
tifications, often establishing them more confidently than he was able to 
do, have also enabled us in many cases to alter and amend his results. 
Such a re-examination was necessary, in order to furnish safe ground for 
a more detailed comparison of the three systems, which, as will be seen 
hereafter, leads to important conclusions respecting their historical rela- 
tions to one another. 

1. Apvint ; this treatise exhibits the form apjini ; in the older lists, 
as also often elsewhere, we have the dual apvin&u, afvayvj&u, "the two 
horsemen, or Acvins." The Acvins are personages in the ancient Hindu 
mythology somewhat nearly corresponding to the Castor and Pollux of 
the Greeks. They arc the divinities of the asterism, which is named 
from them. The group is figured as a horse's head, doubtless in allusion 
to its presiding deities, and not from any imagined resemblance. The 
dual name leads us to expect to find it composed of two stars, and that 
is the number allotted to the asterism by the Qakalya and Khanda- 
Kataka. The Surya-Siddhanta (below, v. 16) designates the northern 
member of the group as its junction-star : that this is the star (? Arietis 
(magn. 3.2), and not a Arietis (magn. 2), as assumed by Colebrooke, is 
shown by the following comparison of positions : 

Acvini .... long., A.D. 660, n° 5o/ . . • . lat o* n'N. 
9 Arietia ... do. 1 3° 56' .... do. 8° 28' N. 

a Arietia ... do. 17 37' .... da 9 57' N. 

Colebrooke was misled in this instance by adopting, for the number 
of stars in the asterism, three, as stated by the later authorities, and then 
applying to the group as thus composed the designation given by our 
text of the relative position of the junction-star as the northern, and he 
accordingly overlooked the very serious error in the determination of 
the longitude thence resulting. Indeed, throughout his comparison, he 
gives too great weight to the determination of latitude, and too little to 
that of longitude : we shall see farther on that the accuracy of the lat- 
ter is, upon the whole, much more to be depended upon than that of the 
former. 

Considered as a group of two stars, AQvinl is composed of and % 
Arietis (magn. 4.3) ; as a group of three, it comprises also « in the same 
constellation. 

There is no discordance among the different authorities examined by 
us as regards the position of the junction-star of Acviui, either in lati- 
tude or in longitude. The case is the same with the 8th, 10th, 12th, 
and 13th asterisms, and with them alone. 

The first Arab manzil is likewise composed of (? and y Arietis, to 
which some add a : it is called ash-Sharatan, " the two tokens* — that i» 
to say, of the opening year. 

The Chinese series of sieu commences, as did anciently the Hindu 
system of asterisms, with that which is later the third asteris**, Tne 



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828 E. Burgess, etc., [viii. 9. 

twenty-seventh sieu, named Leu (M. Biot has omitted to give us the 
signification of these titles), is /? Arietis, the Hindu junction-star. 

2. Bharani; also, as plural, bharanyas; from the root 6Aar, "carry": 
in the Taittiriya lists the form apabharant, " bearer away," in singular 
and plural, is also found. Its divinity is Yama, the ruler of the world 
of departed spirits ; it is figured as the yoni, or pudendum muliebre. 
All authorities agree in assigning it three stars, and the southernmost is 
pointed out below (v. 18) as its junction-star. The group is unquestion- 
ably to be identified with the triangle of faint stars lying north of the 
back of the Ram, or 35, 39, and 41 Arietis : they are figured by some as 
a distinct constellation, under the name of Musca Borealis. The desig- 
nation of the southern as the junction-star is not altogether unambigu- 
ous, as 35 and 41 were, in A. D. 560, very nearly equidistant from the 
equator ; the latter would seem more likely to be the one intended, 
since it is nearer the ecliptic, and the brightest of the group — being of 
the third magnitude, while the other two are of the fourth : the defined 
position, however, agrees better with 35, and the error in longitude, as 
compared with 41, is greater than that of any other star in the series : 

Bharani a4° 35' . . . . n° 6' N. 

85 Arietis (a Musca) . . a6° 54' . . . . u°i7'N. 
41 Arietis (c Musca) . . a8° io' . . . . io° 26' N. 

The Graha-Laghava gives Bharani 1° more of polar longitude : this 
would reduce by the same amount the error in the determination of its 
longitude by the other authorities. 

The second Arab manzil, al-Butain, " the little belly" — L e., of the 
Ram — is by most authorities defined as comprising the three stars in the 
haunch of the Ram, or e, J, and q 3 (or else t) Arietis. Some, however, 
have regarded it as the same with Musca ; and we cannot but think that 
al-Blrun!, in identifying, as he does, Bharani with al-Butain, meant to 
indicate by the latter name the group of which the Hindu asterism is 
actually composed. 

The last Chinese sieu, Oei, is the star 35 Arietis, or o Muscae. 

3. Krltika ; or, as plural, krttihas : the appellative meaning of the 
word is doubtful. The regent of the asterism is Agni, the god of fire. 
The group, composed of six stars, is that known to us as the Pleiades. 
It is figured by some as a flame, doubtless in allusion to its presiding 
divinity : the more usual representation of it is a razor, and in the choice 
of this symbol is to be recognized the influence of the etymology of the 
name, which may be derived from the root Jcart y " cut ;" in the configur- 
ation of the group, too, may be seen, by a sufficiently prosaic eye, a 
broad-bladed knife, with a short handle. If the designation given below 
(v. 18) of the southern member of the group as its junction-star, be 
strictly true, this is not Alcyone, or y Tauri (magn. 3), the brightest of 
the six, but either Atlas (27 Tauri : magn. 4) or Merope (23 Tauri : 
magn. 5): the two latter were very nearly equally distant from the 
equator of A. D. 560, but Atlas is a little nearer to the ecliptic. The 
defined position agrees best with Alcyone, nor can we hesitate to regard 
this as actually the junction-star of the asterism. We compare the posi- 
tions below : 



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viii. 9.] S&rya-Siddhdnta. 829 

Krttika 39 8' 4° 44' N. 

Alcyone . . . . 3 9 ° 58' 4° i' N. 

21 Tauri . . • . 4o° ao' 3° 53' N. 

23 Tauri . . . . 39 4i' 3° 55' N. 

The Siddhanta-Qiromani etc. give Krttika 2' leas of polar longitude 
than the Surya-Siddhanta, and the Graha-Laghava, on the other hand, 
30' more : the latter, with the Khanda-Kataka, agree with our text as 
regards the polar latitude, which the others reckon at 4° 30', instead of 5°. 

The Pleiades constitute the third manzil of the Arabs, which is de- 
nominated ath-Thuraiya, " the little thick-set group," or an-Najm, " the 
constellation." Alcyone is likewise the first Chinese sieu, which is 
•tyled Mao. 

4. Rohint, li ruddy" ; so named from the hue of its principal star. 
Prajapati, " the lord of created beings," is the divinity of the asterism. 
It contains five stars, in the grouping of which Hindu fancy has seen the 
figure of a wain (compare v. 13, below) ; some, however, figure it as a 
temple. The constellation is the well-known one in the face of Taurus 
to which we give the name of the Hyades, containing », d, y, #, a Tauri ; 
the latter, the most easterly (v. 19) and the brightest of the group — 
being the brilliant star of the first magnitude known as Aldebaran — is 
the junction-star, as is shown by the annexed comparison of positions : 

Rohini 48° 9' .... 4° 49' a 

Aldebaran ... 49° 45' .... 5° 3o' S. 

The Siddhanta-Qiromani etc. here again present the insignificant vari- 
ation from the polar longitude of our text, of 2' less : the former also 
makes its polar latitude 4° 30' : the Graha-Laghava reads, for the polar 
longitude, 49°. All these variations add to the error of defined position. 

The fourth Arab manzil is composed of the Hyades : its name is ad- 
Dabaran, "the follower" — i. e., of the Pleiades. We would suggest the 
inquiry whether this name may not be taken as an indication that the 
Arab system of mansions once began, like the Chinese, and like the 
Hindu system originally, with the Pleiades. There is, certainly, no very 
obvious propriety in naming any but the second of a series the u follow- 
ing" (sequent or secundus). Modern astronomy has retained the title as 
that of the principal star in the group, to which alone it was often also 
applied by the Arabs. 

The second Chinese sieu, Pi, is the northernmost member of the same 
group, />r e Tauri, a star of the third to fourth magnitude. 

5. Mrgapinha^ or mrgafiras, " antelope's head" : with this name the 
figure assigned to the asterism corresponds : the reason for the designa- 
tion we have not been able to discover. Its divinity is Soma, or the 
moon. It contains three stars, of which the northern (v. 16) is the 
determinative. These three can be no other than the faint cluster in the 
head of Orion, or i, $1, q> 2 Ononis, although the Hindu measurement 
of the position of the junction-star, 1 (magn. 4), is far from accurate, 
especially as regards its latitude : 

Mfgactaha . . . . 6i° 3' . . . . 9 49' S. 
X Ononis .... 63° 4o' ... . i3° 25' S. 



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830 E. Burgess, etc., [viiL 0. 

In this erroneous determination of the latitude all authorities agree : 
the Graha-Laghava adds 1° to the error in polar longitude, reading 62° 
instead of 63°. 

Here again there is an entire harmony among the three systems com- 
pared. The Arab munzil, al-Hak'ah, is composed of the same stars 
which make up the Hindu asterism : the third sieu, named Tse, is the 
Hindu junction-star, i Orionis. 

6. Ardrd, " moist :" the appellation very probably has some meteoro- 
logical ground, which we have not traced out : this is indicated also by 
the choice of Rudra, the storm-god, as regent of the asterism. It com- 
prises a single star only, and is figured as a gem. It is impossible not to 
regard the bright star of the first magnitude in Orion's right shoulder, 
or a Orionis, as the one here meant to be designated, notwithstanding 
the very grave errors in the definition of its position given by our text : 
the only visible star of which the situation at all nearly answers to that 
definition is 135 Tauri, of the sixth magnitude; we add its position 
below, with that of a Orionis : 

ArdriL 65° 5o' . . . . 8° 53' S. 

a Orionis . ... 68° 43' .... i6° 4' S. 
135 Tauri . . . 67 38' . . . . 9 io' S. 

The distance from the sun at which the heliacal rising and setting of 
Ardra is stated below (ix. 14) to take place would indicate a star of about 
the third magnitude ; this adds to the difficulty of its identification with 
cither of the two stars compared. We confess ourselves unable to 
account for the confusion existing with regard to this asterism, of which 
al-Blruni also could obtain no intelligible account from his Indian 
teachers. But it is to be observed that all the authorities, excepting our 
text and the ^akalya-SanhitA, give Ardra 11° of polar latitude instead 
of 9°, which would reduce the error of latitude, as compared with a 
Orionis, to an amount very little greater than will be met with in one or 
two other cases below, where the star is situated south of the ecliptic ; 
and it is contrary to all the analogies of the system that a faint star 
should have been selected to form by itself an asterism. The Siddhanta- 
£iroraani etc. make the polar longitude of the asterism 20' less than that 
given by the Surya-Siddhanta, and the Graha-Laghava 1° 20' less : these 
would add so much to the error of longitude. 

Here, for the first time, the three systems which we are comparing 
disagree with one another entirely. The Chinese have adopted for the 
determinative of their fourth sieu, which is styled Tsan, the upper star 
in Orion's belt, or d Orionis (2) — a strange and arbitrary selection, for 
which M. Biot is unable to find any explanation. The Arabs have estab- 
lished their sixth station close to the ecliptic, in the feet of Pollux, nam- 
ing it al-Han'ah, " the pile" : it comprises the two stars y (2.3) and £ 
(4.3) Geminorum : some authorities, however, extend the limits of the 
mansion so far as to include also the stars in the foot of the other twin, 
or 17, v, (i Geminorum ; of which the latter is the next Chinese sieu. 

7. Punarvasu ; in all the more ancient lists the name appears as a 
dual, punarvasft : it is derived from punar, "again," and vasu, "good, 
brilliant" : the reason of the designation is not apparent The regent 



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Tiii. 9.] SCLrt/a-Siddhdnta. 831 

of the asterism is Aditi, the mother of the Adityas. Its dual title indi- 
cates that it is composed of two stars, of nearly equal brilliancy, and 
two is the number allotted to it by the £&ka1ya and Khanda-Kataka, the 
eastern being pointed out below (v. 19) as the junction-star. The pair 
are the two bright stars iu the heads of the Twins, or a and |? Gemino- 
rum, and the latter (1.2) is the junction-star. The comparison of posi- 
tions is as follows : 

Punarvasu .... 9a 5a' .... 6° o' N. 

/3 Geminorum. . . 93° id' . . . . 6° 39' N. 

The Graha-Laghava adds 1° to the polar longitude of Punarvasu as 
stated by the other authorities. 

Four stars are by some assigned to this asterism, and with that num- 
ber corresponds the representation of its arrangement by the figure of 
a house : it is quite uncertain which of the neighboring stars of the same 
constellation are to be added to those above mentioned to form the group 
of four, but we think * (magn. 4) and v (5) those most likely to have 
been chosen : Colebrooke suggests # (3.4) and t (5.4). 

The determinative of the fifth sieu, Tsing, is p Geminorum (3), which, 
as we have seen, is reckoned among the stars composing the sixth 
manzil : the seventh xnanzil includes, like the Hindu asterism, a and p 
Geminorum : it is named adh-Dhira', " the paw" — i. e., of the Lion ; the 
figure of Leo (see Ideler, p. 152 etc.) being by the Arabs so stretched 
out as to cover parts of Gemini, Cancer, Canis Minor, and other neigh- 
boring constellations. 

8. Pushya ; from the root push, " nourish, thrive" ; another frequent 
name, which is the one employed by our treatise, is tishya, which is 
translated "auspicious"; Amara gives also sidhya, " prosperous." Its 
divinity is Brhaspati, the priest and teacher of the gods. It comprises 
three stars — the Khanda-Kataka alone seems to give it but one — of which 
the middle one is the junction-star of the asterism. This is shown by 
the position assigned to it to be d Cancri (4) : 

Pushya .... 106 o' . . . . o° o' 
6 Cancri .... 108 4a' .... o° 4' N. 

£he other two are doubtless y (4.5) and & (6) of the same constellation : 
the asterism is figured as a crescent and as an arrow, and the arrange* 
ment of the group admits of being regarded as representing a crescent, 
or the barbed head of an arrow. Were the arrow the only figure given, 
it might be possible to regard the group as composed of y, #, and (? (4), 
the latter representing the head of the arrow, and the nebulous cluster, 
Praesepe, between y and #, the feathering of its shaft: # (105° 43' — 
0° 48' S.) would then be the junction-star. 

The Arab rnanzil, an-Nathrah, " the nose-gap" — i. e., of the Lion — 
comprises y and 8 Cancri, together with Praesepe ; or, according to somo 
authorities, Praesepe alone. The sixth sieu, Kuei, is # Cancri, a star 
whteh is, at present, only with difficulty distinguished by the naked eye. 
Ptolemy rates it as of the fourth magnitude, like y and 8 : perhaps it is 
one of the stars of which the brilliancy has sensibly diminished during 
the past two or three thousand years, or else a variable star of very 
long period. The possibility of such changes requires to be taken into 
account, in comparing our heavens with those of so remote a past 



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332 R Burgess, etc., [viii. 9. 

0. Afleshd; or, as plural, dfleskds; the word is also written dfreshd: 
its appellative meaning is " entwiner, embracer." With the name accord 
the divinities to whom the regency of the asterism is assigned, which 
are sarpds, the serpents. The number of stars in the group is stated as 
five by all the authorities excepting the Khanda-Kataka, which reads six r 
their configuration is represented by a wheel. The^tar a Cancri (4) is 
pointed out by Colebrooke as the junction-star of Aclesha, apparently 
from the near correspondence of its latitude with that assigned to the 
latter, for he says nothing in connection with it of his native helpers : 
but a Cancri is not the eastern (v. 19) member of any group of five stars ; 
nor, indeed, is it a member of any distinct group at all. Now the name* 
figure, and divinity of Ac,lesba> are all distinctive, and point to a constel- 
lation of a bent or circular form : and if we go a little farther south- 
ward from the ecliptic, we find precisely such a constellation, and one- 
containing, moreover, the corresponding Chinese determinative. The 
group is that in the head of Hydra, or ?, <r, tf, e, q Hydras, o and q being 
of the fifth magnitude, and the rest of the fourth : their arrangement is 
conspicuously circular. There can be no doubt, therefore, that the 
situation of the asterism is in the head of Hydra, and e Hydro, its 
brightest star (being rated in the Greenw. Cat as of magnitude 3.4, 
while 8 is 4.5), is the junction-star : 

AclesbA .... 109 59' ... . 6° 56' S. 
• Hydra. . . . na° 20' . . . . 11 8' S. 
a Cancri .... n3° 5' . . . . 5° 3i' S. 

The error of the Hindu determination of the latitude is, indeed, very 
considerable, yet not greater than we are compelled to accept in one or 
two other cases. The Khanda-Kataka increases it 1°, giving the aster- 
ism 6° instead of 1° of polar latitude. The Siddhanta-Qiromani etc. 
deduct 1° from the polar longitude of the Surya-Siddhanta, and the 
Graba-Laghava deducts 2° : both variations would add to the error in 
longitude. 

The Arab manzil is, in this instance, far removed from the Hindu aster- 
ism, being composed of £ Cancri (5) and I Leonis (5.4), and called at- 
Tarf, " the look" — i. e., of the Lion. The seventh Chinese sieu, Lieu, is, 
as already noticed, included in the Hindu group, being $ Hydra?. 

10. MaghA ; or, as plural, magh&s; " mighty." The pitaras. Fathers, 
or manes of the departed, are the regents of the asterism, which is fig- 
ured as a house. It is, according to most authorities, composed of five 
stars, of which the southern (v. 18) is the junction-star. Four of these 
must be the bright stars in the neck and side of the Lion, or £, y, 7. and 
a Leonis, of magnitudes 4.5, 2, 3.4, and 1.2 respectively ; but which 
should be the fifth is not easy to determine, for there is no other single 
star which seems to form naturally a member of the same group with 
these: v (5), n (5), or 9 (4) might be forced into a connection with 
them. This difficulty would be removed by adopting, with the Khanjla- 
Kataka, six as the number of stars included in the asterism : it would 
then be composed of all the stars forming the conspicuous constellation 
familiarly known as " the Sickle." The star o Leonis, or Regulus, the 
most brilliant of the group, is the junction-star, and its position is defined 
with unusual precision : 



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viii. 9.] S&rya-Siddh&nta. 

Magha .... 1290 o' .... o° o' 
Regulus .... 129 49' .... o° 27' N. 

The tenth manzil, aj-Jabhah, " the forehead" — i. e., of the Lion — is 
also composed of £, y, V, a Leonis. 

The eighth, ninth, and tenth sieu of the Chinese system altogether 
disagree in position with the groups marking the Hindu and Arab man* 
•ions, being situated far to the southward of the ecliptic, in proximity, 
according to Biot, to the equator of the period when they were estab- 
lished, fhe eighth, Sing, is a Hydrae (2), having longitude (A. D. 560) 
127° 16', latitude 22° 26' S. 

11, 12. Phalguni; or, as plural, phalgunyas; the dual, phalguny&u, 
is also found : this treatise presents the derivative form ph&lfjunt, which 
is not infrequently employed elsewhere. The word is likewise used to 
designate a species of tig-tree : its derivation, and its meaning, as applied 
to the asterisms, is unknown to us. Here, as in two other instances, 
later (the 20th and 21st, and the 26th and 27th asterisms), we have 
two groups called by the same name, and distinguished from one another 
qa purva and uttara, "former" and "latter" — that is to say, coming ear- 
lier and later to their meridian-transit. The true original character and 
composition of these three double asterisms has been, if we are not mis- 
taken, not a little altered and obscured in the description of them fur- 
nished to us ; owing, apparently, to the ignorance or carelessness of the 
describers, and especially to their not having clearly distinguished the 
characteristics of the combined constellation from those of "its separate 
parts. In each case, a couch or bedstead (pay yd, maflca^ paryanka) is 
given as the figure of one or both of the parts, and we recognize in 
them all the common characteristic of a constellation of four stars, form- 
ing together a regular oblong figure, which admits of being represent- 
ed — not unsuitably, if rather prosaically — by a bed. This figure, in the 
case of the Phalgunis, is composed of tf, #, ft and 93 Leonis, a very 
distinct and well-marked constellation, containing two stars, d and ft of 
the second to third magnitude, one, #, of the third, and one, 03, of the 
fourth. The symbol of a bed, properly belonging to the whole constel- 
lation, is given by all the authorities to both the two parts into which it 
is divided. Each of these latter has two stars assigned to it, and the 
junction-stars are said (v. 18) to be the northern. The first group is, 
then, clearly identifiable as d and # Leonis, the former and brighter 
being the distinctive star : 

Panra-Phalgun! . . . i39° 58' . . . . ii° 19' N. 

5 Leonis i4i° i5' . . . . i4° 19' N. 

S Leonis i43° 24' . . . . 9 4o' N. 

The Siddhanta-Qiromani etc., and the Graha-Laghava, give Purva- 
Phalgunl respectively 3° and 4° more of polar longitude than the Surya- 
Siddhanta. These are more notable variations than are found in any 
other case, and they appear to us to indicate that these treatises intend to 
designate #, the southern member of the group, as its junction-star : wo 
have accordingly added its position also above. 

In the latter group, the junction-star is evidently § Leonis : 
Uttara-Phalgunl . . . i5o° 10' . . . . ia° 5' N. 

Leonis i5i° 37' . . . . 12 17' tf. 

vox* vi. 43 



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884 JK Jfrrg&s, efc, [tiiL ft. 

• This star, however, is not the northern, hut the southern, of the two 
composing Jjbe asterism : its description as the southern we cannot hut 
regard as simply an error, founded on a misapprehension of the compo- 
sition of the double group. To al-Biruni, Leonis and another star to 
the northward, in the Arab constellation Coma Berenices, were pointed 
out as forming the asterism Uttara-Phalguni. The Qakalya gives it five 
stars, probably adding to Leonis the four small stars in the head of the 
Virgin, $ l 9 f, jt, and ©, of magnitudes four to five and five. . , - 

The regents of Purva and Uttara-Phalguni are Bhaga and Aryamajfc 
or Aryaman and Bhaga, two of the Adityas. 

The two corresponding Arab mansions are called az-Zubrah, "the 
mane" — L e., of the Lion — and as-Sarfoh, "the turn": they agree as 
nearly as possible with the Hindu asterisms, the former being composed 
of d and # Leonis, the latter of (? Leonis alone. The Chinese ««, named 
respectively Chang and Y, are v* Hydras (5)* and « Crateris (4). 

1 8. Hti9ta, " hand." Savitar, the sun, is regent of the asterism, whieh, 
in accordance with its name, is figured as a hand, and contains five stars, 
corresponding to the five fingers. These are the five principal stars in 
the constellation Coitus, a well-marked group, which bears, however, 
no very conspicuous resemblance to a hand. The stars are named- 
counting from the thumb around to the little finger, according to our 
apprehension of the figure — ft «, *, y, and d Corvi. The text gives be- 
low (v. 17) a very special description of the situation of the junction- 
star in the group, but one which is unfortunately quite hard to under- 
stand and apply : we regard it as most probable, however (see note to 
y. 17), that y (3) is the star intended : the defined position, in which ail 
the authorities agree, would point rather to 8 (3) : 

Hasta ...... 174 22' ... . io° 6 f S. 

y Com .... 170 44' . . . . i4° 29' S. 

$ Corvi . , . . 173° 27' ... . ia° io'S. 

The Hindu and Chinese systems return, in this asterism, to an accord- 
ance with one another : the eleventh sieu f Chin, is the star / Corvi. The 
Arab system holds its own independent course one point farther : its 
thirteenth mansion comprises the five bright stars & 7, /, #, * Virginia, 
which form two sides, measuring about 15° each, of a great triangle: 
the mansion is named al- Auwa', " the barking dog." 

14. Citrd, "brilliant" This is the beautiful star of the first magni- 
tude a Virginis, or Spica, constituting an asterism by itself, and figured 
as a pearl or as a lamp. Its divinity is Tvashtar, " the shaper, artificer." 
Its longitude is very erroneously denned by the Surya-Siddbanta : 
Citra . . . . 180 48' . , . . i° 5o' S. 
Spica .... i83° 49' . • » : a° a' S. 

All the other authorities, however, saving the $akalya, remove this 
error, by giving Citra 183° of polar longitude, instead of 180°. The 
only variation from the definition of latitude made by our text is offered 
by the Siddhanta-£ironiani, which, varying for once from the Brahma- 
Siddhanta, reads 1° 45' instead of 2°. 

• It is, apparently, by an original error of the press, that M. Biot, in all his 
tables, calk tbia star »'. :_-..:., 



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44iii0j S&ryar&iddkdnta. 835 

- Spica is likewise the fourteenth manzil of the Arabs, styled by them 
*s-Simak, and the twelfth *ieu of the Chinese, who call it Kio. 

15. Svdity or tv&ti; the word is said to mean " sword." The Taifc- 
tirlya-Brahmana calls the asterism nishtyd, " outcast," possibly from its 
remote northern situation. It is, like the last, an asterism comprising 
but a single brilliant star, which is figured as a coral bead, gem, or peari* 
In the definition of its latitude all authorities agree ; the Graba-Laghava 
makes its polar longitude 198° only, instead of 199°. The star intended 
k plainly * Bootis, or Arcturus : 

SvAtl i83° a' .... 33° 5o' N. 

Arcturus. . . • i84° 12' . . . . 3o° 57' N. 

In this instance, the Hindus have gone far beyond the limits of the 
fodiac, in order to bring into their series of asterisms a brilliant star 
from the northern heavens: the other two systems agree in remaining 
near the ecliptic The ftfiiifonnfli Chinese steu, Kang, is * Virginis 
(4.5) ; the Arab manzil, al-Ghafr, " the covering/ 1 includes the same 
star, together with *, and either I or <p Virginis. 

16* Vif&kkd, " having spreading branches" : in all the earlier lists the 
name appears as a dual, vif&khe. The asterism is also placed under the 
regency of a dual divinity, indr&gnt, Indra and Agni. We should ex- 
pect, then, to find it composed, like the other two dual asterisms, the 1st 
and 7th, of two stars, nearly equal in brilliancy, and two is actually the 
number assigned to the group by the £akalya and the Kbanda-Kataka. 
Now the only two stars in this region of the zodiac forming a conspicur 
6ns .pair are a and § Libras, both of the second magnitude, and as these 
two compose the corresponding Arab mansion, while the former of them 
is the Chinese sieu, we have the strongest reasons for supposing them to 
constitute the Hindu asterism also. There are, however, difficulties in 
the way of this assumption. The later authorities give Vic&kha four 
starst and the defined position of the junction-star identifies it neither 
with <* nor ft but with the faint star * (4.3) in the the same constella- 
tion. Colebrooke, overlooking this star, suggests a or * Libra (5) : the 
following comparison of positions will show that neither of them can be 
the one meant to be pointed out : 



VicikU . . . 


. 3l3° k 3>' . . 


. . i°a5'S. 


t Libra . . . 


. 2X1° O' . . 


. . i°4» r SL 


a Libra . . . 


. ao5° 5' . . 


. . o° a3' X. 


« Libra . » • 


. ai7°45' - - 


. . o° a' N. 



The group is figured as a torana ; this word Jones and Colebrooke 
translate " festoon," but its more proper meaning is " an outer door or 
gate, a decorated gateway." And if we change the designation of situ- 
ation of the junction-star in its group, given below (v. 16); from " north- 
ern" to u southern," we find without difficulty a quadrangle of stars, viz. 
*» a » ft V (4-$) Libr®, which admits very well of being figured as a gater 
way. Nor is it, in our opinion; taking an unwarrantable liberty to make 
such an alteration. The whole scheme of designations we regard as 
of inferior authenticity, and as partaking of the confusion and uncer- 
tainty of the later knowledge of the Hindus respecting their system of 
asterisms. That they were long ago doubtful of the position of V ic&kk& 



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SS6 B. Burgess, ete, [viiL 9. 

k shown by the feet thai al-Birunt was obliged to nark it in bis list as 
" unknown/ 9 Very probably the Surya-Siddhanta, in calling * the north- 
ern member of the group, intended to include with it only the star 20 
Librae (3.4), situated about 6° to the south of it. Upon the whofo, then, 
while we regard the identification of VicaJtha as in some respects mora 
doubtful than that of any other asterism in the series, we yet believe 
that it was originally composed of the two stars a and p Librae, and that 
later the group was extended to include also * and y, and, as so extended, 
was figured as a gateway. The selection, contrary to general usage, of 
the faintest star in the group as its junction-star, may have been made 
jn order to insure against the reversion of the asterism to its original 
dual form. 

The variations of the other authorities from the position as stated in 
our text are of small importance: the Siddhanta-^iromani etc. give 
Vicakha 55' less of polar longitude, and the Graha-Laghava 1° less; of 
polar latitude, the Siddhanta-^iromani gives it 10', the Graha-Laghava 
30' less ; the Khanda-Kataka agrees here, as also in the two following 
asterisms, with the Surya-Siddhanta. 

The sixteenth Arab manzil, comprising, as already noticed, a and (? 
Librae, is styled as-Zubanan, " the two claws" — L c, of the Scorpion : 
the name of the corresponding Chinese mansion, having for its deter- 
minative « Librae, is Ti. 

17. AnurddhA ; or, as plural, anurddkds .^ the word meano " success." 
The divinity is Mitra, "friend" one of the Adityas. According to the 
Qakalya, the asterism is composed of three stars, and with this onr text 
plainly agrees, by designating (v. 18) the middle as the jiraction*sta? : 
all the other authorities give it four stars. As a group of throe, it com- 
prises ft d, n Scorpionis, 8 (2.3) being the junction-star ; as the fourth 
member wc are doubtless to add ? Scorpionis (5.4). It is figured as a 
bali or vali ; this Colebrooke translates " a row of oblations" ; we do 
not find, however, that the word, although it means both " oblation, 
offering," and u a row, fold, ridge," is used to designate the two com- 
bined : perhaps it may better be taken as simply " a row ;" the stars 
of the asterism, whether considered as three or four, being disposed in 
nearly a straight line. The comparison of positions is as follows : 

AnurtoM .... «4° 44' .... a° 5a' S. 
8 Scorpionis . . . 222 34' . . . . " i° 57' S. 

The Siddhanta-Qiromani and Graha-Laghava estimate the latitude of 
Anuradha somewhat more accurately, deducting from the polar latitude, 
as given by our text, 1° 15' and 1° respectively : the Siddh&nta-Qiromani 
etc. also add the insignificant amount of 5' to the polar longitude of the 
Surya-Siddhanta. 

The corresponding Arab manzil, named al-Iklil, " the crown," con* 
tains also the three stars ft 6, n Scorpionis, some authorities adding ? to 
the group. The Chinese sieu, Fang, is n (3), the southernmost and .the 
faintest of the three. 

18. Jyeshthd, " oldest." The Taittiriya-Sanhita, in its list of aster* 
isms, repeats here the name rohxtii^ " ruddy," which we have had above 
as that of the 4th asterism : the appellation has the same ground in this 



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viiL 9.] Sbry&Siddh&nta. 887 

as in the other case, the junction-star of Jyeshtha being also one of those 
which shine with a reddish light. The regent is Indra, the god of the 
clear sky. The group contains, according to all the authorities, three 
stars, and the central one (v. 1 8) is the junction-star. This is the bril- 
liant star of the first magnitude a Scorpionis, or Antares ; its two com* 
panions are <r (3.4) and t (3.4) in the same constellation : 

Jyeshtha .... 23o° 7' .... 3° 5o' S. 
Antares .... 229 44' .... 4° 3i' S. 

The constellation is figured as a ring, or ear-ring ; by this may be un- 
derstood, perhaps, a pendent ear-jewel, as the three stars of Jyeshtha 
form nearly a straight line, with the brightest in the middle. 

The Siddhanta-^iromani and Graha-Laghava add to the polar longi- 
tude of the junction-star of the asterism, as stated in our text, 5' and 1° 
respectively, and they deduct from its polar latitude 30' and 1° respect- 
ively, making the definition of its position in both respects less accurate. 

Antares forms the eighteenth manxil, and is styled al-Kalb, "the 
gfeart" — i. e., of the Scorpion : or and 1 are called an-Niyat, u the prch 
J0rat0." The Chinese «cw, Sin, is the westernmost of the three, or o\ 

19. Mbla, " root" The presiding divinity of the asterism is ntrr/i, 
u calamity," who is also regent of the south-western quarter. It com- 
prises, according to the £akalya, nine stars ; their configuration is rep- 
resented by a lion's tail. The stars intended are those in the tail of the 
Scormon, or *, f* f £, ?, #, *, x, v, X Scorpionis, all of them of the third, 
or tnrd to fourth, magnitude. Other authorities count eleven stars in 
the group, probably reckoning fi and £ as four stars ; each being, in feet, 
a group of two closely approximate stars, named in our catalogues ft * 
(3), ft* (4), K l (4.5), £2 (3). The Khanda-Kataka alone gives Mula only 
two stars, which are identified by al-Blronl with the Arab manzil ash- 
Shaulah, or I and v Scorpionis. The Taittirlya-Sanhita, too, gjves the 
name of the asterism as vicrt&u, 4 * the two releasers" : the Vicrtau are 
several times spoken of in the Atharva-Veda as two stars of which the 
rising promotes relief from lingering disease (kshetriya) : it is accord- 
ingly probable that these are tic two stars in the sting of the Scorpion, 
and that they alone have been regarded by some as composing the aster- 
ism : their healing virtue would doubtless be connected with the meteor- 
ological conditions of the time at which their heliacal rising takes place. 
Our text (v. 19) designates the eastern member of the group as its junc- 
tion-star: it is uncertain whether the direction is meant to apply to the 
group of two, or to that of nine stars : if, as seems probable, A is the 
star pointed out by the definition of position, it is strictly true only of 
the pair A and v, since *, *, and # are all farther eastward than I : 

Mula a4a° 5a' ... . 8° 48' S. 

X Scorpionis . . . a44° 53' . ... i3° 44' S. 

The Graha-Laghava gives a more accurate statement of the longitude, 
adding 1° to the polar longitude as defined by all the other authorities : 
but it increases the error in latitude, by deducting 1° from that presented 
by our text : the Siddbanta-Qiromani, in like manner, deducts 30', while 
the Khanda-Kataka adds the same amount 



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8S8 R Bwffes*, eic^ [riiw ft 

The T&ittirly*SaiihitA makes pitoras, the Fathers, the presiding divftp. 
ities of this asterism, as well as of the tenth. - 

Bentley states (Hind. Astr., p. 5) that Mula was originally reckoned as 
the first of the asterisms, and was therefore so named, as being their root 
or origin ; also that, at another time, or in a different system, the series 
was made to begin with Jyeshth&, which thence received its title of 
" eldest" These statements are put forth with characteristic reckless- 
ness, and apparently, like a great many others in his pretended history 
of Hindu astronomy, upon tne unsupported authority of his own conjee-; 
ture. It is, in many cases, by no means easy to discover reasons for the 
particular appellations by which the asterisms are designated : but wo 
would suggest that Mula may perhaps have been so named from its be- 
ing considerably the lowest, or farthest to the southward, of the whole 
series of asterisms, and hence capable of being looked upon as the root 
out of which they had grown up the heavens. It would even be possi- 
ble to trace the same conception farther, and to regard JyeshthA as so 
styled because it was the first, or " oldest," outgrowth from this root, 
while the Vi$&khe, "the two diverging branches," were the stars ig^ 
which the series broke into two lines, the one proceeding northward, ijJP 
Svktl or Arcturus, the other westward, to Citrft, or Spica. We throw 
out the conjecture for what it may be worth, not being ourselves at alt 
confident of its accordance with the truth. 

The nineteenth Arab manzit is styled ash-Shaulah, "the sting" — L e^" 
of the Scorpion — and comprises, as already noticed, v and k Scortipnis. 
The determinative of the seventeenth sieu, Uei, is included in the Hindu 
asterism, being ft 2 Scorpionis. 

20,21. Ask&dhd; or, as plural, ash&dhfa; this treatise presents the 
derivative form Ashddhd, which is not infrequent elsewhere : the word 
means " unsubdued." Here, again, we have a double group, divided 
into two asterisms, which are distinguished as pkrva and uttara, " former 
and latter." Their respective divinities are &pa* r " the waters," and vipvi 
dev&s, u the collective gods." Two stars are ordinarily allotted to each 
asterism, and in each case the northern is designated (v. 16) as the junc- 
tion-star. By some authorities each group is figured as a bed or couch ; 
by others, the one ad a bed and the other as an elephant's tusk ; and 
here, again, there is a difference of opinion as to which is the bed and 
which the tusk. The true solution of this confusion is, as we conceive, 
that the two asterisms taken together are figured as a bed, while either 
of them alone is represented by an elephant's tusk. The former group 
must comprise 8 (3.4) and * (3.2) Sagittarii, the former being the junc- 
tion-star ; this is shown by the following comparison of positions : . 

PArva-AshAdhA .... a54° 39' .... 5° 28' S. 
5 Sagittarii a54° 3a' .... 6° 25' S. 

The Graba-L&ghava gives Purva-Ash&dhft, 1° more of polar longitude, 
and 30' less of polar latitude, than the Surya*Siddh&nta : the Siddh&nta- 
Qiroraani etc. give it 10' less of the latter. 

The latter of the two groups contains, as its southern star, £ Sagittarii 
(3.4), and its northern and junction-star can be no other than <r (2.3) in 
the samo constellation, notwithstanding the error in the Hindu determP 



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▼iiL^] S&rya-Siddh&nia. £80 

nation of its latkrtde, which led Cotebrooke to regard t (4.3) at the 
star intended : we subjoin the positions : 

TTttara-Ashadha . ... 260° 9? .... 4° 5q' S. 

* Sagittarii ...... 262° 21' .... 3° a4' 8. 

* Sagittttrii a64° 48' .... 5 6 1' S. 

The only variation from the position of the junction-star of this aster- 
ism as stated in our text is presented by the Graha-L&ghava, which 
makes its polar longitude 261° instead of 260°. 
. The £akalya (according to Colebrooke : our MS. is defective at this 
point} and the Khanda-Kataka assign four stars to each of the Ash&dhas, 
and the former represents each as a bed. It would not be difficult to 
establish two four-sided figures in this region of the constellation Sagit- 
tarius, each including the stars above mentioned, with two others : the 
one would be composed of y 2 (4.3), d t e f rj (4 — the star is also called £ 
Telescopii), the other of 9 (4.3), or, t, and £ : such is unquestionably the 
constitution of the two asterisms, considered as groups of four stars ; 
they are thus identified also, it may be remarked, by al-Blrunl. The 
junotion-stars would still be 8 and c, which are the northernmost in their 
respective constellations ; nor is there any question as to which four 
among the eight are selected to make up the double asterism, since J, e, 
C t and or both form the most regular quadrangular figure, and are the 
brightest stars. 

. The determinatives of the eighteenth and nineteenth mansions of tho 
Chinese, Ki and Teu, are y 2 and q> Sagittarii, which are included in the 
two quadruple groups as stated above. The twentieth manzil compre- 
hends all the eight stars which we have mentioned, and is styled an- 
Na'aira, " the pasturing cattle" : some also understand each group of 
four as representing an ostrich, na'ara. The twenty-first manzil, on the 
other hand, al-Baldah, " the town," is described as a vacant space above 
the head of Sagittarius, bounded by faint stars, among which the most 
conspicuous is n Sagittarii (4.5). 

22. Abhijit, " conquering." The regent of the asterism is Brahma. 
The position assigned to its junction-star, which is described as the 
brightest (v. 19) in a group of three, identifies it with a Lyra?, or Vega, 
a star which is exceeded in brilliancy by only one or two others in the 
heavens : 

Abhijit .... a64° jo' . . . . 5q° 58' N. 
Vega .... a65° i5' . . . . 6i° 46' N. 

The other authorities compared (excepting the Qakalya) define tho 
position in latitude of Abhijit more accurately, adding 2° to the polar 
latitude given by the Surya-Siddhanta: the Graha-Laghava also improves 
the position in longitude by adding 1° 20', while the Siddhanta-Qiromani 
etc increase the error by deducting 1° 40'. 

The TAittirlya-Sanhita, (iv. 4. 10) omits Abhijit from its list of the as- 
terisins : the probable reason of its omission in some authorities, or in 
certain connections, and its retention in others, we shall discuss far* 
$her on. 

Abhijit is figured as a triangle, or as the triangular not of the ftng&ta, 
&ft.*qKatie plant; this very distinctly represents the grouping of « Lyre 



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840 R Burgess, efc., [nil 9. 

with the two other fainter stars of the same constellation, a and £, both 
of the fifth magnitude. 

In this and the two following asterisms — as once before, in the fifteenth 
of the series — the Hindus have gone far from the zodiac, in order to 
bring into their system brilliant stars from the northern heavens, while 
the Chinese and the Arab systems agree in remaining in the immediate 
neighborhood of the ecliptic. The twentieth sieu is named Nieu, and 
is the star P Capricorni (3), situated in the head of the Goat : the twen- 
ty-second manzil, Sa'd adh-Dhabih, " felicity of the sacrificer," contain* 
the same star, the group being o (composed of two stars, each of mag- 
nitude 3.4) and /? Capricorni. 

23. Qravaria, "hearing, ear"; from the root p*w, "hear" : another 
name for the asterism, frond, found occurring in the Taittirfya lists, is 
perhaps from the same root, but the word means also " lame." Qravana 
comprises three 6tars, of which the middle one (v. 18) is the junction- 
star : they are to be found in the back and neck of the Eagle, namely 
as y, a, and |? Aquilae ; «, the determinative, is a star of the first to sec- 
ond magnitude, while y and j? are of the third and fourth respectively : 

Crarana .... 282 29' ... . 29 54' N. 
oAquike . . . 28i°4»' .... 29 n'N. 

All the authorities agree as to the polar latitude of Crarana : the 
Siddh&nta-^iromani etc. give it 2° less of polar longitude than our trea- 
tise, and the Graha-Laghava even as much as 5° less. 

The regent of the asterism is Vishnu, and its figure or symbol corrcs- 

Eonds therewith, being three footsteps, representatives of the three steps 
y which Vishnu is said, iu the early Hindu mythology, to have strode 
through heaven. The £akalya, however, gives a trident as the figure 
belonging to fravana. Possibly the name is to be regarded as indica- 
ting that it was originally figured as an ear. 

The Chinese sieu corresponding in rank with (Jravana is called Nfl, 
and is the faint star e Aquarii (4.3). The Arab manzil Sa'd Bnla', 
" felicity of a devourer," or al-Bula', " the devourer," etc., includes the 
same star, being composed of e, ft (4.5), v (5) Aquarii, or, according to 
others, of e and 7 (6) Aquarii, or of p and v. 

24. QravishthA; the word is a superlative formation from the same 
root from which came the name of the preceding asterism, and means, 
probably, " most famous." Another and hardly less frequent appella- 
tion is dhanishthb, an irregular superlative from dJianin, " wealthy." The 
class of deities known as the vasus, " bright, good," are the regents of 
the asterism. It comprises four stars, or, according to the Qakalya and 
Kbanda-Kataka, five : the former, which is given by so early a list as 
that of the Taittiriya-Brahmana, is doubtless the original number. The 
group is the conspicuous one in the head of the Dolphin, composed of 
p, «, y, 8 Delphini, all of them stars of the third, or third to fourth, mag- 
nitude, and closely disposed in diamond or lozenge-form : they are fig- 
ured by the Hindus as a drum or tabor. The junction-star, which is the 
western (v. 17), is £: 

^ravishtba .... 2960 5' .... 35° 33' a 
Delphini .... 296 19' ... . 3i* Sf 8. 



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viii. 9.] S&rya-Siddhdnta. 341 

The only variation from the position assigned in our text to the junc- 
tion-star of Qravishtha, is presented by the Graha-Laghava, which gives 
it 286°, instead of 290°, of polar longitude. Perhaps its intention is to 
point out £ (5) as the junction-star : this is doubtless the one added to 
the other four, on account of its close proximity to them, to make up the 
group of five ; it Res only about half a degree westward from /?. 

The name of the twenty-fourth manzil, Sa'd as-Su'ud, " felicity of 
felicities" — i. e., " most felicitous" — exhibits an accordance with that of 
the Hindu asterism which possibly is not accidental. The two are, how- 
ever, as already noticed, far removed in position from one another, the 
Arab mansion being composed of the two stars ft (3) and I (5.4), in the 
left shoulder of Aquarius, to which some add also 46, or c 1 , Capricorni 
(6). The corresponding sieu, Hili, is the first of them, or |9 Aquarii. 

25. Qatabhishaji "having a hundred physicians" : the form fatobkiskd, 
which seems to be merely a corruption of the other, also occurs in later 
writings. It is, as we should expect from the title, said to be composed 
of a hundred stars, of which the brightest (v. 19) is the junction-star. 
This, from its defined position, can only be A Aquarii (4) : 

^atabhishaj .... 3iq° 5i' . . . . o° 29' S. 
31 Aquarii 32i° 33' .... o° a3' S. 

The rest of the asterism is to be sought among the yet fainter stars in 
the knee of Aquarius, and the stream from his jar : of course, the num- 
ber one hundred is not to be taken as an exact one, nor are we to sup- 
pose it possible to trace out with any distinctness the figure assigned 
to the group, which is a circle. The Khanda-Kataka, according to al- 
Biruni, gives Qatabhishaj only a single star, but this is probably an error 
of the Arab traveller : he is unable to point out which of the stars in 
Aquarius is to be regarded as constituting the asterism. 

The regent of the 25th asterism, according to nearly all the authori- 
ties, is Varuna, the chief of the Adityas, but later the god of the waters : 
the Taittirlya-Sanhita alone (rives to it and to the 14th asterism, as well 
as to the 18th, Indra as presiding divinity: this is perhaps mere blun- 
dering. 

The Graha-Laghava places the junction-star of 5 a tabhishaj precisely 
on the ecliptic ; the Siddhanta-Qironiani etc. give it 20', instead of 30' f 
of polar latitude south. 

The corresponding lunar mansion of the Arabs, Sa'd al-Akhbiyah, 
" the felicity of tents," comprises the three stars in the right wrist and 
hand of the Water-bearer, or y (3), £ (4), 9 (4) Aquarii, together with a 
fourth, which Ideler supposes to be re (5). Since, however, the twenty- 
third Chiuese determinative, Goei, is « Aquarii (3), a star so near as 
readily to be brought into the same group with the other three, we are 
inclined to regard it as altogether probable that the mansion was, at 
least originally, composed of a, p, £, and »/. 

26, 27. Bhddrapadd; as plural, bkddrapadds : also bhadrapadd ; from 
bkodra, " beautiful, happy," and pada, " foot." Another frequent appel- 
lation is proshtkapadd : proshtha is said to mean u carp" and " ox" ; the 
latter signification might perhaps apply here. We have here, once more, 
a double asterism, divided into two parts, which are distinguished from 



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842 R Burgess, etc., [viii. 9. 

one another as p&rva and uttara, "former" and "latter." All authori- 
ties agree in assigning two stars to each of the two groups ; but there is 
not the same accordance as regards the figures by which they are rep- 
resented : by some the one, by others the other, is called a conch or bed, 
the alternate one, in either case, being pronounced a bi-faced figure : the 
Muhurta-Cint&mani calls the first a bed, and the second twins. It ad- 
mits, we apprehend, of little or no question that the Bh&drapad&s are 
properly the four bright stars ft ft, y Pegasi, and « Andromeato — all of 
them commonly reckoned as of the second magnitude — which form 
together a nearly perfect square, with sides measuring about 15° : the 
constellation, a very conspicuous one, is familiarly known as the "Square 
of Pegasus." The figure of a couch or bed, then, belongs, as in the 
case of the other two double asterisms, already explained, to the whole 
constellation, and not to either of the two separate asterisms into which 
it is divided, while, on the other hand, either of these latter is properly 
enough symbolized by a pair of twins, or by a figure with a double 
face. The appropriateness of the designation " feet, found as a part of 
both the names of the whole constellation, is also sufficiently evident, if 
we regarcl the group as thus composed. The junction-star of the former 
half-asterism is, by its defined position, clearly shown to be a Pegasi : 

Purva-Bhadrapadft .... 334° a5' . . . . 22° 3o' N. 
a Pegasi 333° 27' . . . . 19 a5' N. 

The Graha-L&ghava gives the junction-star 1° less of polar longitude, 
which would bring its position to a yet closer accordance, in respect to 
longitude, with a Pegasi : the error in latitude, which is common to all 
the authorities, is not greater than we have met with several times else- 
where. But we are told below (v. 1 6) that the principal star of each of 
these asterisms is the northern, and this would exclude Pegasi alto- 
gether, bringing in as the other member of the first pair some more 
southern star, perhaps £ Pegasi (3.4). The confusion is not less marked, 
although of another character, in the case of the second asterism : in 
the definition of position of its junction-star we find a longitude given 
which is that of one member of the group, and a latitude which is that 
of the other, as is shown by the following comparison : 

Uttara-BbAdrapada .... 347° 

7 Pegasi . 34o° 

a Andromeda 354° 

If we accept either of these two stars as the one of which the posi- 
tion is meant to be defined, we shall be obliged to admit an error in the 
determination either of its longitude or of its latitude considerably 
greater than we have met with elsewhere. Nor is the matter mended 
by any of the other authorities : the only variation from the data of our 
tjxt is presented by the Graha-L&ghava, which reads, as the polar lati- 
tude of Uttara-Bm\drapad&, 27° instead of 26°. There can be no 
doubt that the two stars recognized as composing the asterism are y Pe- 
gasi and a Andromeda, but there has evidently been a blundering con- 
fusion of the two ill making out the definition of position of the juno- 
tion-star. We would suggest the following as a po v sslbhe explanation" of 



16' . . 


. . 24° i'N. 


8' . . 


. . i2°35'N. 


17' . • 


. . 25°4i'N. 



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wi.,9,] S&rya-SwUlh&nta. 348 

this confusion : that originally a and y Pegasi were designated and de- 
scribed as junction-stars of the two half-groups, of which they wero 
respectively the southern members ; that afterward, for some reason — 
perhaps owing to the astrological theory (see above, vii. 21) of the supe- 
riority of a northern star — the rank of junction-star was sought to be 
transferred from the southern to the northern stars of both asterisms ; 
that, in making the transfer, the original constitution of the former 
group was neglected, while in the latter the attempt was made to define 
the real position of the northern star, but by simply adding to the polar 
latitude already stated for y Pegasi, without altering its polar longitude 
also. Al-Birunl, it should be remarked, was unable to obtain from his 
Hindu informants any satisfactory identification of either of these aster- 
isms, and marks both in his catalogue as " unknown." 

The view we have taken of the true character of the two Bhadrapa- 
das is powerfully supported by their comparison with the corresponding 
members of the other two systems. The twenty-sixth and twenty- 
seventh manzils, al-Fargh al-Mu^dim and al-Fargh al-Mukhir, " the fore 
*nd hind spouts of the water-jar," comprise respectively a and (I Pegasi, 
and y Pegasi and a Andromedae ; the determinatives of the twenty- 
fourth and twenty-fifth sieu, Che and Pi, are a and y Pegasi. 

T?he regents of these two asterisms are aja ekapdt and ahi budhtya^ 
the u one-footed goat" and the " bottom-snake," two mythical figures, of 
obscure significance, from the Vedic pantheon. 

28. Revati, " wealthy, abundant." Its presiding divinity is Pushan, 
u the prosperer," one of the Adityas. It is said to contain thirty-two 
stars, which are figured, like those of fravishtha, ^y a j ram or tabor; 
but it would be in vain to attempt to point out precisely the thirty-two 
-which are intended, or to discover in their arrangement any resemblance 
to the figure chosen to represent it. The junction-star of the group is 
said (v, 18) to be its southernmost member : all authorities agree in 
placing it upon the ecliptic, and all excepting our treatise and the 
Qakalya make its position exactly mark the initial point of the fixed 
sidereal sphere. The star intended is, as we have already often had 
occasion to notice, the faint star £ Piscium, of about the fifth magnitude, 
situated in the band which connects the two Fishes. It is indeed very 
near to the ecliptic, having only 13' of south latitude. It coincided in 
longitude with the vernal equinox in the year 572 of out era. 

At the time of al-Blrunl's visit to India, the Hindus seem to have 
been already unable to point out distinctly and with confidence the sit- 
uation in the heavens of that most important point from which they 
held that the motions of the planets commenced at the creation, and at 
which, at successive intervals, their universal conjunction would again 
take place ; for he is obliged to mark the asterism as not certainly iden- 
tifiable. He also assigns to it, as to £atabhishaj, onfy a single star. 

The twenty-sixth Chinese sieu, Koei, is marked by £ Andromedae (4), 
which is situated only 35' east in longitude from £ Piscium, but which 
has 17° 36' of north latitude. The last manzil, Batn al-Hut, u the fish's 
belly," or ar-Risha, 4| the band," seems intended to include the'stars com- 
posing the northern Fish, and with them probably the Chinese deter- 
minative also : but it is extended so far northward as to take in the bright 



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344 



JE Burgess, etc., 



[viii. 9. 



star Andromeda (2), and to this star alone the name of the mansion 
is sometimes applied, although its situation, so far from the ecliptic (in 
lat. 25° 56' N.), renders it by no means suited to become the distinctive 
star of one of the series of lunar stations. 

We present, in the annexed table, a general conspectus of the cones* 
pondences of the three systems ; and, in order to bring out those corres- 
pondences in the fullest manner possible, we have made the comparison 
in three different ways : noting, in the first place, the cases in which the 
three agree with one another ; then those in which each agrees with one 
of the others ; and finally, those in which each agrees with either the 
one or the other of the remaining two. 



Correspondences of the Hindu, Arab, and Chinese Systems of Asterisms. 






Hinda 








Hindu 


Arab 


Ctiioeee 




Hindu 


wilh 


Hindu 


Hindu 


Arab 


with 


with 


with 


No. 


Num. 


Arab 


with 


with 


with 


Arab 


Hindu 


Hindu 




and 
Chinese. 


Arab. 


Chinene. 


Chinese. 
I 


or 

Chinese. 


or 
Chinese. 


or 
Arab. 


i 


Acvini, 


I 


I 


I 


1 


1 


I 


a 


Bharani, 


2* 


a* 


a 


2» 


a 


2* 


2 


3 


Krttika, 


3 


3 


3 


3 


3 


3 


3 


4 Rohini 


4 


4 


4 


4 


4 


4 


4 


5 


Mrgacirsha, 


5 


5 


5 


5 


5 


5 


5 


6 


Ardra, 


.. 






t 


.. 


.. 


.. 


7 


Punanrasu, 




6 




t 


6 


6 


t 


8 


Pushya, 


6t 


7 


6 




7 


7 


6 


9 


Acleshft, 


.. 


.. 


7 


.. 


8 


• • 


7 


10 


Magfaa, 


.. 


8 


.. 


.. 


9 


8 


•• 


ii 


P.-Phalguni, 


.. 


9 


.. 


.. 


IO 


9 


• • 


i2 j U.-Phalguni, 


.. 


IO 


.. 


.. 


ii 


10 


.. 


i3 


Hasta, 


.. 


.. 


8 


.. 


12 


• • 


8 


i4 


Citra, 


7 


ii 


9 


6 


i3 


11 


9 


i5 


Svati, 






.. 


7 


.. 


12 


10 


16 


VicakhA, 


8 


12 


IO 


8 


i4 


i3 


11 


17 


Anaradha, 


9 


i3 


ii 


9 


i5 


i4 


12 


?8 


Jyeshtha, 


IO 


i4 


la 


IO 


16 


i5 


13 


»9 


Mala,' 


"J 


i5 


l3 




17 


16 


i4 


ao 


P.-Aahadha, 


12 


16 


i4 


ii 


i8 


17 


i5 


ai 


U.-Asbadha, 


.. 


.. 


i5 


.. 


19 




16 


22 ' Abhijit, 


.. 




.• 


12 


.. 


18 


17 • 


23 (^ravana, 


.. 


.. 


.. 


i3 


.. 


19 


18 


24 


^ravishthft, 


.. 


.. 


.. 


r4 


.. 


20 


19 


a5 


(^atabhishaj, 


.. 


.. 


.. 


i5 


.. 


21 


20 


36 


P.-Bhadrapada, 


i3 


17 


16 


16 


20 


22 


21 


27 


U.-Bhadrapada, 


i4 


18 


17 


17 


21 


23 


22 


28 


Revati, 


•• 


•• 


•• 


i8§ 


•• 


*4§ 


*3§ 



* This supposes the second manzil to be composed of the stars in Musca, as 
defined by some authorities. \ The sixth manzil includes, according to many 
authorities, the fifth sieu, but as there is, at any rate, a discordance in the order of 
succession, we hare not reckoned this among the correspondences. \ We reckon 
these two as cases of general coincidence, because,* although the Chinese tiett h not 
contained in the Arab mansion, the Hindu asterism includes them both, and the 
yirtual correspondence of the three systems is beyond dispute. § Here we assume 
the Chinese *ieu to be comprised among the stars forming the last manzil, which is 
altogether probable, although nowhere distinctly stated. 



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viil 0.] Shrya-SiddJi&nta. 345 

Owing to the different constitution of the systems, their correspond- 
ences are somewhat diverse in character : we account the Hindu aster- 
kins and the Arab mansions to agree, when the groups which mark the 
two are composed, in whole or in part, of the same stars : we account 
the Chinese system to agree with the others, when the determinative of 
a sieu is to be found among the stars composing their groups. We 
have prefixed to the whole the numbers and titles of the Hindu aster- 
isms, for the sake of easy reference back to the preceding detailed iden- 
tifications and comparisons. 

After this exhibition of the concordances existing among the three 
systems, it can, we apprehend, enter into the mind of no one to doubt 
that all have a common origin, and are but different forms of one and 
the same system. The questions next arise — is either of the three the 
original from which the others have been derived \ and if so, which of 
them is entitled to the honor of being so regarded ? and aTe the other 
two independent and direct derivatives from it, or does either of them 
come from the other, or must both acknowledge an intermediate source f 
In endeavoring to answer these questions, we will first exhibit the views 
of M. Biot respecting the origin and character of the Chinese sieu, as 
stated in the volumes for 1840 and 1859 of the Journal des Savants. 

According to Biot, the sieu form an organic and integral part of that 
system by which the Chinese, from an almost immemorial antiquity, 
have been accustomed to make their careful and industrious observations 
of celestial phenomena. Their instruments, and their methods of ob- 
servation, have been closely analogous with those in use among modern 
astronomers in the West: they have employed a meridian-circle and a 
measure of time, the clepsydra, and have observed meridian-transits, ob- 
taining right ascensions and decimations of the bodies observed. To 
reduce the errors of their imperfect time-keepers, they long ago selected 
certain stars near the equator, of which they determined with great care 
the intervals in time, and to these they referred the positions of stars or 
planets coming to the meridian between them. The stars thus chosen 
are the sieu. Twenty-four of them were fixed upon more than two 
thousand years before our era (M. Biot says, about B. C. 2357 : but it is 
obviously impossible to fix the date, by internal evidence, within a cen- 
tury or two, ncrr is the external evidence of a more definite character) ; 
the considerations which governed their selection were three : proximity 
to the equator of that period, distinct visibility — conspicuous brilliancy 
not being demanded for them — and near agreement in respect to time 
of transit with the upper and lower meridian-passages of the bright stars 
near the pole, within the circle of perpetual apparition : M. Biot finds 
reason to believe that these circumpolar stars had been earlier observed 
with special care, and made standards of comparison, and that, when 
it was afterward seen to be desirable to have stations near the equator, 
such stars were adopted as most nearly agreed with them in right ascen- 
sion. The other four, being the 8th, 14th, 21st, and 28th, the accession 
of which completed the system of twenty -eight, were added in the time 
of Cheu-Kong, about B.C. 1100, because they marked very nearly the 
positions of the equinoxes and solstices at that epoch : the bright star of 
the Pleiades, however, which had originally been made the first of the 



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$46 & Burgess, etc t [vjii. ft, 

series, from its near approach to the vernal equinox of that remoter era, 
still maintained, as it has ever since maintained, its rank as the first. 
Since the time of Che u- Kong the system has undergone no farther modi- 
fication, bat has been preserved unaltered and unimproved, with the 
obstinate persistency so characteristic of the Chinese, although many of 
the determinative stars have, under the influence of the precession, be- 
come far removed from the equator, one of them even having retro- 
graded into the preceding mansion. 

If the history of the Chinese sieu, as thus drawn out, is well-founded 
and true, the question of origin is already solved: the system of twenty- 
eight celestial mansions it proved to be of native Chinese institution — 
just as the system of representation of the planetary movements by epi- 
cycles is proved to be Greek by the fact that we can trace in the history 
of Greek science the successive steps of its gradual elaboration. That 
history rests, at present, upon the authority of M. Biot alone : we are 
not aware, at least, that any other investigator has gone independently 
over the same ground ; and he has not himself laid before us, in their 
original form, the passages from Chinese texts which furnish the basis 
of his conclusions. But we regard them as entitled to be received, 
upon his authority, with no slight measure of confidence : his own dis- 
tinguished eminence as a physicist and astronomer, his familiarity with 
researches into the history and archaeology of science, his access to the 
abundant material for the history of Chinese astronomy collected and 
worked up by the French missionaries at Pekin, and the zealous assist- 
ance of his son, M. fidouard Biot, the eminent Sinologist, whose prema- 
ture death, in 1 850, has been so deeply deplored as a severe loss to Chi- 
nese studies — all these advantages, rarely united in such fullness in the 
person of any one student of such a subject, give very great weight to 
views arrived at by him as the results of laborious and long-continued 
investigation. Nor do we see that any general considerations of import- 
ance can be brought forward in opposition to those views. It is, in the 
firftt place, by no moans inconsistent with what we know in other res- 
pects of the age and character of the culture of the Chinese, that they 
should have devised such a system at so early a date. They have, from 
the beginning, been as much distinguished by a tendency to observe and 
record as the Hindus by the lack of such a tendency : they have always 
attached extreme importance to astronomical labors, and to the construc- 
tion and rectification of the calendar ; and the industry and accuracy 
of their observations is attested by the use made of them by modern 
astronomers — thus, to take a single instance, of the cometary orbits 
which have been calculated, the first twenty-five rest upon Chinese ob- 
servations alone : and once more, it is altogether in accordance with the 
clever empiricism and practical shrewdness of the Chinese character that 
they should have originated at the very start a system of observation 
exceedingly well adapted to its purpose, stopping with that, working in- 
dustriously on thenceforth in the same beaten track, and never develop- 
ing out of so promising a commencement anything deserving the name 
of a science, never devising a theory of the planetary motions, never 
even recognizing and defining the true character of the cardinal phe- 
nomenon of the precession. 



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viii. 9.] SbryarSiddhdnta. 847 

Again, although it might seem beforehand highly improbable that a 
system of Chinese invention should have found its way into the West, 
and have been extensively accepted there, many centuries before the 
Christian era, there are no so insuperable difficulties in the way as should 
destroy the force of strong presumptive evidence of the truth of such a 
communication. It is well known that in very ancient times the pro- 
ducts of the soil and industry of China were sought as objects of lux- 
ury in the West, and mercantile intercourse opened and maintained 
across the deserts of Central Asia ; it even appears that, as early as 
about B. C. 600 (Isaiah xlix. 12), some knowledge of the Sinim, as a far- 
off eastern nation, had penetrated to Babylon and Judea. On the other 
hand, we do not know how much, if at all, earlier than this it may be 
necessary to acknowledge the system of asterisms to have made its ap- 
pearance in India. The literary memorials of the earliest period, the 
Vedic period proper, present no evidence of the existence of the system : 
indeed, it is remarkable how little notice is taken of the stars by the 
Vedic poets ; even the recognition of some of them as planets does not 
appear to have taken place until considerably later. In the more recent 
portions of the Vedic texts — as in the nineteenth book of the Atharva- 
Veda, a modern appendage to that modern collection, and in parts of 
the YajuT-Veda, of which there is reason to believe that the canon was 
not closed until a comparatively late period — full lists of the asterisms 
are found. The most unequivocal evidence of the early date of the sys- 
tem in India is furnished by the character of the divinities under whose 
regency the several asterisms are placed : these are all from the Vedic 
pantheon ; the popular divinities of later times are not to be found among 
them ; but, on the other hand, more than one whose consequence is lost, 
and whose names almost are forgotten, even in the epic period of Hindu 
history, appear in the list Neither this, however, nor any other evi- 
dence known to us, is sufficient to prove, or even to render strongly prob- 
able, the existence of the asterisms in India at so remote a period that 
the system might not be believed to have been introduced, in its fully 
developed form, from China. 

If, now, we make the attempt to determine, upon internal evidence, 
which of the three systems is the primitive one, a detailed examination 
of their correspondences and differences will lead us first to the import- 
ant negative conclusion that no one among them can be regarded as the 
immediate source from which either of the other two has been derived. 
It is evident that the Hindu asterisms and the Arab man&zil constitute, 
in many respects, one and the same system : both present to us constel- 
lations or groups of stars, in place of the single determinatives of the 
Chinese rieu ; and not only are those groups composed in general of the 
same stars, but in several cases — as the 7th, 10th, 11th, and 12th mem- 
bers of the series — where they differ widely in situation from the Chi- 
nese determinatives, they exhibit an accordance with one another which 
is too close to be plausibly looked upon as accidental. But if it is thus 
made to appear that neither can have come independently of the other 
from a Chinese original, it is no less certain that neither can have come 
through the other from such an original ; for each has its own points of 
agreement with the stew, which the other does not share — the Hindu in 



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848 E. Burgess, etc., [viii. 9. 

the 9th, 13*th, and 21st asterisms, the Arab in the 15th, 22nd, 23rd, 
24th, and 25th mansions. The same considerations show, inversely, that 
the Chinese system cannot be traced to either of the others at its source, 
since it agrees in several points with .each one of them where that one 
differs from the third. It becomes necessary, then, to introduce an addi- 
tional term into the comparison ; to assume the existence of a fourth 
system, differing in some particulars from each of the others, in which 
all shall find their common point of union. Such an assumption is not 
to be looked upon as either gratuitous or arbitrary. Not only do the 
mutual relations of the three systems point distinctly toward it, but it is 
also supported by general considerations, and will, we think, be found to 
remove many of the difficulties which have' embarrassed the history of 
the general system. It has been urged as a powerful objection to the 
Chinese origin of the twenty-eight-fold division of the heavens, that we 
find traces of its existence in so many of the countries of the West, 
geographically remote from China, and in which Chinese influence can 
hardly be supposed to have been directly felt. And it is undoubtedly 
true that neither India nor Arabia has stood in ancient times in such 
relations to China as should fit it to become the immediate recipient 
of Chinese learning, and the means of its communication to surround- 
ing peoples. The great route of intercourse between China and the 
West led over the table-land of Central Asia, and into the north- 
eastern territory of Iran, the seat of the Zoroastrian religion and cul- 
ture : thence the roads diverged, the one leading westward, the other 
south-eastward into India, through the valley of the Cabul, the true 
gate of the Indian peninsula. Within or upon the limits of this central 
land of Iran we conceive the system of mansions to have received that 
form of which the Hindu nakshatras and the Arab man&zil are the 
somewhat altered representatives : precisely where, and whether in the 
hands of Semitic or of Aryan races, we would not at present attempt 
to say. There are, as has been noticed above, traces of an Iranian sys- 
tem to be found in the Bundehesh ; but this is a work which, although 
probably not later than the times of Persia's independence under her 
Sassanian rulers, can pretend to no high antiquity, and no like traces have 
as yet been pointed out in the earliest Iranian memorial, the Zendavcsta. 
Weber (Ind. Literaturgeschichte, p. 221), on the other hand, sees in the 
mazzaloth and mazzarotk of the Scriptures (Job xxxviii. 82 ; II Kings 
xxiii. 5) — words radically akin with the Arabic manzil — indications of 
the early existence of the system in question among the western Semites, 
and suspects for it a Chaldaic origin : but the allusions appear to us too 
obscure and equivocal to be relied upon as proof of this, nor is it easy 
to believe that such a method of division of the heavens should have 
prevailed so far to the west, and from so ancient a time, without our 
hearing of it from the Greeks ; and especially, if it formed a part of the 
Chaldaic astronomy. This point, however, may fairly be passed over, 
as one to be determined, perhaps, by future investigations, and not of 
essential importance to the present inquiry. The question of originality 
is at least definitely settled adversely to the claims of both the Hindu 
and the Arab systems, and can only lie between the Chinese and that 
fourth system from which the other two have together descended. And 



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▼HL 9.] S&rya-Siddh&nta. 849 

as concerns these, we are willing to accept the solution which is fur- 
nished us by the researches of M. Biot, supported as we conceive it to be 
by the general probabilities of the case. Any one who will trace out, 
by the help of a celestial globe or map,* the positions of the Chinese 
determinatives, cannot fail to perceive their general approach to a great 
circle of the sphere which is independent of the ecliptic, and which 
accords more nearly with the equator of B. 0. 2850 than with any other 
later one. The full explanations and tables of positions given by Biot 
(Joura. d. Sav., 1840, pp. 243-254) also furnish evidence, of a kind ap- 
preciable by all, that the system may have had the origin which ho 
attributes to it, and that, allowing for the limitations imposed upon it by 
its history, it is consistent with itseli and well enough adapted to the 
purposes for which it was designed. * With the positions of its determin- 
ative stars seem to have agreed those of the constellations adopted by 
the common parent of the Hindu and Arab systems, excepting in five or 
six points : those points being where the Chinese make their one unac- 
countable leap from the head to the belt of Orion, and again, where the 
steu are drawn off far to the southward, in the constellations Hydra and 
Crater : and this, in our view, looks much more as if the series of the 
sieu were the original, whose guidance had been closely followed except- 
ing in a few cases, than as if the asterisms composing the other systems 
had been independently selected from the groups of stars situated along 
the zodiac, with the intention of forming a zodiacal series. It is easy to 
see, farther, how the single determinatives of the $iett should have become 
the nuclei for constellations such as are presented by the other systems; 
but i£ on the contrary, the sieu had been selected by the Chinese, in 
each case, from groups previously constituted, there appears no reason 
why their brightest stars should not have been chosen, as they were cho- 
sen later by the Hindus, in the establishment of junction-stars for the 
asterisms. 

We would suggest, then, as the theory best supported by all the evi- 
dence thus far elicited, that a knowledge of the Chinese astronomy, and 
with it the Chinese system of division of the heavens into twenty-eight 
mansions, was carried into Western Asia at a period not much later 
than B. C. 1100, and was there adopted by some western people, either 
Semitic or Iranian. That in their hands it received a new form, such as 
adapted it to a ruder and less scientific method of observation, the limit- 
ing stars of the mansions being converted into zodiacal groups or con- 
stellations, and in some instances altered in position, so as to be brought 
nearer to the general planetary path of the ecliptic. That in this 
changed form, having become a means of roughly determining and de- 
scribing the places and movements of the planets, it passed into the 
keeping of the Hindus— very probably along with the first knowledge 
of the planets themselves — and entered upon an 'independent career of 
history in India. That it still maintained itself in its old seat, leaving its 
traces later in the Bundehesh ; and that it made its way so far westward 
as finally to become known to, and adopted by, the Arabs. The farther 

* We propose to famish at the close of this publication, in connection -with the 
additional notes, such a map of the zodiacal sone of the heavens as will sufficiently 
illustrate the character and mutual relations of (he three systems compared. 
vol. vi. 45 



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850 E. Burgess, etc., [viii. 9. 

modifications introduced into it by the latter people all hare in view a 
single purpose, that of establishing its stations in the immediate neigh* 
borhood of the ecliptic : to this purpose the whole Arab system is not 
less constantly faithful than is the Chinese to its own guiding principle. 
The Hindu sustains in thi6 respect but an unfavorable comparison with 
the others : the arbitrary introduction, in the 15th, 22nd, 23rd, and 24th 
asterisms, of remote northern stars, greatly impairs its unity, and also 
furnishes an additional argument of no slight force against its original- 
ity ; for, on the one hand, the derivation of the others from it becomes 
thereby vastly more difficult, and, on the other, we can hardly believe 
that a system of organic Indian growth could have become disfigured in 
India by such inconsistencies ; they wear the aspect, Tather, of arbitrary 
alterations made, at the time of itsradoption, in an institution imported 
from abroad. 

It might, at first sight, appear that the adoption by the Arabs of the 
manzil corresponding to Acvinl as the first of their series indicated that 
they had derived it from India posterior to the transfer by the Hindus 
of the first rank from Krttika, the first of the sieu, to Acvinl : but the 
circumstance seems readily to admit of another interpretation. The 
names of many of the Arab mansions show the influence of the Greek 
astronomy, being derived from the Greek constellations : the same influ- 
ence would fully explain an arrangement which made the series begin 
with the group coinciding most nearly with the beginning of the Greek 
zodiac. The transfer on the part of the Hindus, likewise, was unques- 
tionably made at the time of the general reconstruction of their astro- 
nomical system under the influence of western science. The two series 
are thus to be regarded as having been brought into accordance in this 
respect by the separate and independent working of the same cause. 

M. Biot insists strongly, as a proof of the non-originality of the sys- 
tem of asterisms among the Hindus, upon its gross and palpable lack 
of adaptedness to the purpose for which they used it ; he compares it 
to a gimlet out of which they have tried to make a saw. In this view 
we can by no means agree with him : we would rather liken it to a 
hatchet, which, with its edge dulled and broken, has been turned and 
made to do duty as a hammer, and which is not ill suited to its new and 
coarser office. Indeed, taking the Hindu system in its more perfect 
and consistent form, as applied by the Arabs," and comparing it with the 
Chinese rieu at any time within the past two thousand years, we are by 
no means sure that the advantage in respect to adaptation would not be 
generally pronounced to be upon the side of the former. The distance 
of many of the sieu during that period from the equator, the faintness 
of some among them, the great irregularity of their intervals, render 
them anything but a ^ model system for measuring distances in right 
ascension. On the otner hand, to adopt a series of conspicuous constel- 
lations along the zodiac, by their proximity to which the movements of 
the planets shall be marked, is no unmotived proceeding : just such a 
division of the ecliptic among twelve constellations preceded and led the 
way to the Greek method of measuring by signs, having exact limits, 
and independent of the groups of stars which originally gave name to 
them. M. Biot's error lies in his misapprehension, m two important 



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viil 9.] S&rya-Siddh&nta. 851 

respects, of the character of the Hindu asterisms : in the first place, he 
constantly treats them as if they were, like the sieu, single stars, the in- 
tervals between whose circles of declination constituted the accepted 
divisions of the zodiac ; and in the second place, he assumes them to 
have been established for the purpose of marking the moon's daily pro- 
gress from point to point along the ecliptic. Now, as regards the first 
of these points, we hare already shown above that the conversion of the 
Chinese determinatives into constellations took place, in all probability, 
before their introduction to the knowledge of tne Hindus : there is, in- 
deed, an entire unanimity of evidence to the effect that the Hindu sys- 
tem is from its inception one of groups of stars: this is conclusively 
shown by the original dual and plural names of the asterisms, or by their 
otherwise significant titles— compare especially those of the 13th and 
25th of the series. The selection of a "junction-star" to represent the 
asterism appears to be something comparatively modern : we regard it 
as posterior to the reconstruction of the Hindu astronomy upon a truly 
scientific basis, and the determination, by calculation, of the precise pla- 
ces of the planets : this would naturally awaken a desire for, and lead 
to, a similarly exact determination of the position of some star repre- 
senting each asterism, which might be employed in the calculation of 
conjunctions, for astrological purposes; the astronomical uses of the 
system being no longer of much account after the division of the ecliptic 
into signs. And the choice of the junction-star has fallen, in the ma- 
jority of cases, not upon the Chinese determinative itself, but upon some 
other and more conspicuous member of the group originally formed 
about the latter. Again, there is an entire absence of evidence that the 
" portions" of the asterisms, or the arcs of the ecliptic named from them, 
were ever measured from junction-star to junction-star : whatever may 
be the discordance among the different authorities respecting their extent 
and limits, they are always freely, and often arbitrarily, taken from parts 
of the ecliptic adjacent to, or not far removed from, the successive con- 
stellations. 

As regards the other point noticed, it is, indeed, not at all to be won- 
dered at that M. Biot should treat the Hindu nakthatras as a system 
bearing special relations to the moon, since, by those who have treated 
of them, they have always been styled u houses of the moon," u moon- 
stations," " lunar asterisms," and the like. Nevertheless, these designa- 
tions seem to be founded only in carelessness, or in misapprehension. 
In the Surya-Siddhanta, certainly, there is no hint to be discovered of 
any particular connection between them and the moon, and for this rea- 
son we have been careful never to translate the term nakshatra by any 
other word than simply " asterism." Nor does the case appear to have 
been otherwise from the beginning. No one of the general names for 
the asterisms (nakshatra, bha, dhishnya) means literally anything more 
than " star" or " constellation" : their most ancient and usual appella- 
tion, nakshatra, is a word of doubtful etymology (it may be radically 
akin with nakta, nox, rf(, " night"), but it is not infrequently met with 
in the Vedic writings, with the general signification of "star," or 
" group of stars" : the moon is several times designated as " sovereign 
of the nakshatras" but evidently in no other sense than that in which 



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852 R Burgess, etc., [viii. 9* 

we style her " queen of night" ; for the same title is in other passages 
given to the sun, and even also to the Milky Way. When the name 
came to be especially applied to the system of zodiacal asterisms, we 
have seen above that a single one of the series, the 5th, was placed un- 
der the regency of the moon, as another, the 13th, under that of the 
snn : this, too, by no means looks as if the -whole design of the system 
was to mark the moon's daily motions. Naturally enough, since the 
moon is the most conspicuous of the nightly luminaries, and her revolu- 
tions more rapid and far more important than those of the othecs, the 
asterisms would practically be brought into much more frequent use in 
connection with ner movements : their number, likowise, being nearly 
accordant with the number of days of her sidereal revolution, could not 
but tempt those who thus employed them to set up an artificial relation 
between the two. Hence the Arabs distinctly call their divisions of the 
zodiac, and the constellations which mark them, " houses of the moon/' 
and, until the researches of M. Biot, no one, so far as we are aware, had 
ever questioned that the number of the asterisms or mansions, wherever 
found, was derived from and dependent on that of the days in the 
moon's revolution. It was most natural, then, that Western scholars, 
having first made acquaintance with the Arab system, should, on finding 
the same in India, call it by the same name : nor is it very strange, even, 
that Ideler should have gone a step farther, and applied the familiar title 
of " lunar stations" to the Chinese sieu also ; an error for which he is 
sharply criticised by M. Biot (Journ. d. Sav., 1859, p. 480). The latter 
cites from al-Biruni (Journ. d. Sav. 1845, p. 49 ; 1859, pp. 487-8) two 
passages derived by him from Varaha-mikira and Brahmagupta respect- 
ively, in which are recorded attempts to establish a systematic relation 
between the asterisms and the moon's true and mean daily motions. 
One of these passages is exceedingly obscure, and both are irreconcila- 
ble with one another, and with what we know of the system of aster- 
isms from other sources : two conclusions, however, bearing upon the 
present matter, are clearly derivable from them : first, that, as the " por- 
tions" assigned to the asterisms had no natural and fixed limits, it was 
possible for any Hindu system-maker so to define them as to bring tliem 
into a connection with the moon's daily motions : and secondly, that 
such a connection was never deemed an essential feature of the system, 
and hence no one form of it was generally recognized and accepted. 
The considerations adduced by us above are, we think, fully sufficient to 
account for any such isolated attempts at the establishment of a con- 
nection as al-Biruni, who naturally sought to find in the Hindu nokskor 
tras the correlatives of his own man&zil al-kamar, was able to discover 
among the works of Hindu astronomers : there is no good reason why 
we should deprive the former of their true character, which is that of 
zodiacal constellations, rudely marking out divisions of the ecliptic, and 
employable for all the purposes for which such a division is demanded. 

The reason of the variation in the number of the asterisms, which are 
reckoned now as twenty-eight and now as twenty-seven, is a point of no 
small difficulty in the history of the system. M. Biot makes the acute 
suggestion that the omission of Abhijit from the series took place be- 
cause the mansion belonging to that asterism was on the point of becom- 



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yiii. $.] S&rya-Siddh&nta. 853 

ing extinguished, the circle of declination of its junction-star being 
brought by the precession to a coincidence with that of the junction-star 
of the preceding asterism about A.D. 972. But it has been shown 
above that M. Biot's view of the nature of a nakihatra — that it is, 
namely, the arc of the ecliptic intercepted between the circles of declina- 
tion of two successive junction-stars—is altogether erroneous : however 
nearly those circles might approach one another, there would still be no 
difficulty in assigning to each asterism its " portion" from the neighbor- 
ing region of the ecliptic Again, this explanation would not account 
for the early date of the omission of Abhijit, which, as already noticed, 
is found wanting in one of the most ancient lists, that of the Taittirtya- 
fianhita. It is to be observed, moreover, that M. Blot, in calculating the 
period of Abhijit's disappearance, has adopted * Sagittarii as the junc- 
tion-star of Uttara-Ashadha, while we have shown above that <r, and 
not t, is to be so regarded : and this substitution would defer until sev- 
eral centuries later the date of coincidence of the two circles of declina- 
tion. According to the Hindu measurements, indeed (see the table of 
positions of the junction-stars, near the beginning of this note), Abhijit 
is farther removed from the preceding asterism, both in polar longitude 
and in right ascension, than are five of the other asterisros from their 
respective predecessors: nor does the Hindu astronomical system ac- 
knowledge or make allowance for the alteration of position of the circles 
of declination under the influence of the precession : their places, as 
data for the calculation of conjunctions, are ostensibly laid down for all 
future time. For these various reasons, M. Biot's explanation is to be 
rejected as insufficient A more satisfactory one, in our opinion, may 
be found in the fact, illustrated above (see Fig. 81, beginning of this 
note), that the asterisms are in general so distributed as to accord quite 
well with a division of the ecliptic into twenty-seven equal portions, 
but not with a division into twenty-eight equal portions; that the 
region where they are too much crowded together is that from the 20th 
to the 28 rd asterism, and that, among those situated in this crowded 
quarter, Abhijit is farthest removed from the ecliptic, and so is more 
easily left out than any of the others, in dividing the ecliptic into por- 
tions. We cannot consider it at all doubtful that Abhijit is as originally 
and truly a part of the system of asterisms as any other constellation 
in the series, which is properly composed of twenty-eight members, and 
not of twenty-seven : the analogy of the other systems, and the fact 
that treatises like this Siddhanta, which reckon only twenty-seven divi- 
sions of the ecliptic, are yet obliged, in treating of the asterisms as con- 
stellations, to regard them as twenty-eight, are conclusive upon this 
point The whole difficulty and source of discordance seems to lie in 
this — how shall there, in any systematic method of division of the eclip- 
tic, be found a place and a portion for a twenty-eighth asterism ? The 
Khanda-Kataka, as cited by al-Biruni — in making out, by a method 
which is altogether irrespective of the actual positions of the asterisms 
with reference to the zodiac, the accordance already referred to between 
their portions and the moon's daily motions — allots to Abhijit so much 
of the ecliptic as is equivalent to the mean motion of the moon during 
the part of a day by which her revolution exceeds twenty-seven days. 



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354 E. Burgess, etc., [viii. 9. 

Others allow it a share in the proper portions of the two neighboring 
asterisms : thus the Muhurta-Mal£, a late work, of date unknown to us, 
says : " the last quarter of Uttara-Ashadha and the first fifteenth of 
Qravana together constitute Abltijit : it is so to bo accounted when 
twenty-eight asterisms are reckoned ; not otherwise." Ordinarily, how- 
ever, the division of the ecliptic into twenty-seven equal " portions" is 
made, and Abhijit is simply passed by in their distribution. After the 
introduction of the modern method of dividing the circle into degrees 
and minutes, this last way of settling the difficulty would obviously re- 
ceive a powerful support, and an increased currency, from the fact that 
a division by twenty-seven grave each portion an even number of min- 
utes, 800, while a division by twenty-eight yielded the awkward and 
Unmanageable quotient 771^. 

Much yet remains to be done, before the history and use of the sys- 
tem of asterisms, as a part of the ancient Hindu astronomy and astrol- 
ogy, shall be fully understood. There is in existence an abundant liter- 
ature, ancient and modern, upon the subject, which will doubtless at 
some time provoke laborious investigation, and repay it with interesting 
results. To us hardly any of that literature is accessible, and only the 
final results of wide-extended and long-continued studies upon it could 
be in place here. We have already allotted to the nakskatras more 
space than to some may seem advisable : our excuse must be the in- 
terest of the history of the system, as part of the ancient history of 
the rise and spread of astronomical science ; the importance attaching 
to the researches of M. Biot, the inadequate attention hitherto paid 
them, and the recent renewal of their discussion in the Journal des Sa- 
vants ; and finally and especially, the fact that in and with the asterisms 
is bound up the whole history of Hindu astronomy, prior to its trans- 
formation under the overpowering influence of western science. In the 
modern astronomy of India, the nakskatras are of subordinate conse- 
quence only, and appear as hardly more than reminiscences of a former 
order of things : from the Surya-Siddhanta might be struck out every 
line referring to them, without serious alteration of the character of the 
treatise.